Another Book of Somos-Like  Miracles

                              By Shalosh B. Ekhad



                                Prop. Number, 1

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                      2
                              a(n - 1) (4 + 5 a(n - 1) )
                       a(n) = --------------------------
                                                 2
                                   2 + 5 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+5*a(n-1)^2)/(2+5*a(n-2)^2)
                       subject to the initial conditions

                              a(1) = 3, a(2) = 21

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 2

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                       2
                             2 a(n - 1) (1 + 3 a(n - 1) )
                      a(n) = ----------------------------
                                                 2
                                   1 + 6 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+3*a(n-1)^2)/(1+6*a(n-2)^2)
                       subject to the initial conditions

                              a(1) = 4, a(2) = 56

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 3

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                       2
                              a(n - 1) (4 + 21 a(n - 1) )
                       a(n) = ---------------------------
                                                  2
                                   2 + 21 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+21*a(n-1)^2)/(2+21*a(n-2)^2)
                       subject to the initial conditions

                              a(1) = 5, a(2) = 115

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 4

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                       2
                             2 a(n - 1) (1 + 8 a(n - 1) )
                      a(n) = ----------------------------
                                                  2
                                   1 + 16 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+8*a(n-1)^2)/(1+16*a(n-2)^2)
                       subject to the initial conditions

                              a(1) = 6, a(2) = 204

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 5

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                       2
                              a(n - 1) (4 + 45 a(n - 1) )
                       a(n) = ---------------------------
                                                  2
                                   2 + 45 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+45*a(n-1)^2)/(2+45*a(n-2)^2)
                       subject to the initial conditions

                              a(1) = 7, a(2) = 329

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 6

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             2 a(n - 1) (1 + 15 a(n - 1) )
                      a(n) = -----------------------------
                                                  2
                                   1 + 30 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+15*a(n-1)^2)/(1+30*a(n-2)^2)
                       subject to the initial conditions

                              a(1) = 8, a(2) = 496

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 7

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                       2
                              a(n - 1) (4 + 77 a(n - 1) )
                       a(n) = ---------------------------
                                                  2
                                   2 + 77 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+77*a(n-1)^2)/(2+77*a(n-2)^2)
                       subject to the initial conditions

                              a(1) = 9, a(2) = 711

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 8

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             2 a(n - 1) (1 + 24 a(n - 1) )
                      a(n) = -----------------------------
                                                  2
                                   1 + 48 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+24*a(n-1)^2)/(1+48*a(n-2)^2)
                       subject to the initial conditions

                             a(1) = 10, a(2) = 980

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 9

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                       2
                             a(n - 1) (4 + 117 a(n - 1) )
                      a(n) = ----------------------------
                                                  2
                                  2 + 117 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+117*a(n-1)^2)/(2+117*a(n-2)^2)
                       subject to the initial conditions

                             a(1) = 11, a(2) = 1309

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 10

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             2 a(n - 1) (1 + 35 a(n - 1) )
                      a(n) = -----------------------------
                                                  2
                                   1 + 70 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+35*a(n-1)^2)/(1+70*a(n-2)^2)
                       subject to the initial conditions

                             a(1) = 12, a(2) = 1704

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 11

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                       2
                             a(n - 1) (4 + 165 a(n - 1) )
                      a(n) = ----------------------------
                                                  2
                                  2 + 165 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+165*a(n-1)^2)/(2+165*a(n-2)^2)
                       subject to the initial conditions

                             a(1) = 13, a(2) = 2171

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 12

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             2 a(n - 1) (1 + 48 a(n - 1) )
                      a(n) = -----------------------------
                                                  2
                                   1 + 96 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+48*a(n-1)^2)/(1+96*a(n-2)^2)
                       subject to the initial conditions

                             a(1) = 14, a(2) = 2716

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 13

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                       2
                             a(n - 1) (4 + 221 a(n - 1) )
                      a(n) = ----------------------------
                                                  2
                                  2 + 221 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+221*a(n-1)^2)/(2+221*a(n-2)^2)
                       subject to the initial conditions

                             a(1) = 15, a(2) = 3345

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 14

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             2 a(n - 1) (1 + 63 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                   1 + 126 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+63*a(n-1)^2)/(1+126*a(n-2)^2)
                       subject to the initial conditions

                             a(1) = 16, a(2) = 4064

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 15

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                       2
                             a(n - 1) (4 + 285 a(n - 1) )
                      a(n) = ----------------------------
                                                  2
                                  2 + 285 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+285*a(n-1)^2)/(2+285*a(n-2)^2)
                       subject to the initial conditions

                             a(1) = 17, a(2) = 4879

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 16

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             2 a(n - 1) (1 + 80 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                   1 + 160 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+80*a(n-1)^2)/(1+160*a(n-2)^2)
                       subject to the initial conditions

                             a(1) = 18, a(2) = 5796

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 17

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                       2
                             a(n - 1) (4 + 357 a(n - 1) )
                      a(n) = ----------------------------
                                                  2
                                  2 + 357 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+357*a(n-1)^2)/(2+357*a(n-2)^2)
                       subject to the initial conditions

                             a(1) = 19, a(2) = 6821

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 18

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             2 a(n - 1) (1 + 99 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                   1 + 198 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+99*a(n-1)^2)/(1+198*a(n-2)^2)
                       subject to the initial conditions

                             a(1) = 20, a(2) = 7960

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 19

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                       2
                             a(n - 1) (4 + 437 a(n - 1) )
                      a(n) = ----------------------------
                                                  2
                                  2 + 437 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+437*a(n-1)^2)/(2+437*a(n-2)^2)
                       subject to the initial conditions

                             a(1) = 21, a(2) = 9219

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 20

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 120 a(n - 1) )
                     a(n) = ------------------------------
                                                  2
                                  1 + 240 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+120*a(n-1)^2)/(1+240*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 22, a(2) = 10604

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 21

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                       2
                             a(n - 1) (4 + 525 a(n - 1) )
                      a(n) = ----------------------------
                                                  2
                                  2 + 525 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+525*a(n-1)^2)/(2+525*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 23, a(2) = 12121

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 22

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 143 a(n - 1) )
                     a(n) = ------------------------------
                                                  2
                                  1 + 286 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+143*a(n-1)^2)/(1+286*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 24, a(2) = 13776

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 23

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                       2
                             a(n - 1) (4 + 621 a(n - 1) )
                      a(n) = ----------------------------
                                                  2
                                  2 + 621 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+621*a(n-1)^2)/(2+621*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 25, a(2) = 15575

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 24

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 168 a(n - 1) )
                     a(n) = ------------------------------
                                                  2
                                  1 + 336 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+168*a(n-1)^2)/(1+336*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 26, a(2) = 17524

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 25

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                       2
                             a(n - 1) (4 + 725 a(n - 1) )
                      a(n) = ----------------------------
                                                  2
                                  2 + 725 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+725*a(n-1)^2)/(2+725*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 27, a(2) = 19629

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 26

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 195 a(n - 1) )
                     a(n) = ------------------------------
                                                  2
                                  1 + 390 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+195*a(n-1)^2)/(1+390*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 28, a(2) = 21896

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 27

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                       2
                             a(n - 1) (4 + 837 a(n - 1) )
                      a(n) = ----------------------------
                                                  2
                                  2 + 837 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+837*a(n-1)^2)/(2+837*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 29, a(2) = 24331

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 28

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 224 a(n - 1) )
                     a(n) = ------------------------------
                                                  2
                                  1 + 448 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+224*a(n-1)^2)/(1+448*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 30, a(2) = 26940

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 29

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                       2
                             a(n - 1) (4 + 957 a(n - 1) )
                      a(n) = ----------------------------
                                                  2
                                  2 + 957 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+957*a(n-1)^2)/(2+957*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 31, a(2) = 29729

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 30

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 255 a(n - 1) )
                     a(n) = ------------------------------
                                                  2
                                  1 + 510 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+255*a(n-1)^2)/(1+510*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 32, a(2) = 32704

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 31

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 1085 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 1085 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+1085*a(n-1)^2)/(2+1085*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 33, a(2) = 35871

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 32

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 288 a(n - 1) )
                     a(n) = ------------------------------
                                                  2
                                  1 + 576 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+288*a(n-1)^2)/(1+576*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 34, a(2) = 39236

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 33

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 1221 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 1221 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+1221*a(n-1)^2)/(2+1221*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 35, a(2) = 42805

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 34

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 323 a(n - 1) )
                     a(n) = ------------------------------
                                                  2
                                  1 + 646 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+323*a(n-1)^2)/(1+646*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 36, a(2) = 46584

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 35

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 1365 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 1365 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+1365*a(n-1)^2)/(2+1365*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 37, a(2) = 50579

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 36

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 360 a(n - 1) )
                     a(n) = ------------------------------
                                                  2
                                  1 + 720 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+360*a(n-1)^2)/(1+720*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 38, a(2) = 54796

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 37

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 1517 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 1517 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+1517*a(n-1)^2)/(2+1517*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 39, a(2) = 59241

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 38

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 399 a(n - 1) )
                     a(n) = ------------------------------
                                                  2
                                  1 + 798 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+399*a(n-1)^2)/(1+798*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 40, a(2) = 63920

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 39

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 1677 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 1677 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+1677*a(n-1)^2)/(2+1677*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 41, a(2) = 68839

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 40

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 440 a(n - 1) )
                     a(n) = ------------------------------
                                                  2
                                  1 + 880 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+440*a(n-1)^2)/(1+880*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 42, a(2) = 74004

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 41

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 1845 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 1845 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+1845*a(n-1)^2)/(2+1845*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 43, a(2) = 79421

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 42

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 483 a(n - 1) )
                     a(n) = ------------------------------
                                                  2
                                  1 + 966 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+483*a(n-1)^2)/(1+966*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 44, a(2) = 85096

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 43

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 2021 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 2021 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+2021*a(n-1)^2)/(2+2021*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 45, a(2) = 91035

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 44

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 528 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                  1 + 1056 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+528*a(n-1)^2)/(1+1056*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 46, a(2) = 97244

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 45

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 2205 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 2205 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+2205*a(n-1)^2)/(2+2205*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 47, a(2) = 103729

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 46

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 575 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                  1 + 1150 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+575*a(n-1)^2)/(1+1150*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 48, a(2) = 110496

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 47

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 2397 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 2397 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+2397*a(n-1)^2)/(2+2397*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 49, a(2) = 117551

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 48

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 624 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                  1 + 1248 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+624*a(n-1)^2)/(1+1248*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 50, a(2) = 124900

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 49

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 2597 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 2597 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+2597*a(n-1)^2)/(2+2597*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 51, a(2) = 132549

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 50

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 675 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                  1 + 1350 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+675*a(n-1)^2)/(1+1350*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 52, a(2) = 140504

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 51

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 2805 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 2805 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+2805*a(n-1)^2)/(2+2805*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 53, a(2) = 148771

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 52

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 728 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                  1 + 1456 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+728*a(n-1)^2)/(1+1456*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 54, a(2) = 157356

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 53

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 3021 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 3021 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+3021*a(n-1)^2)/(2+3021*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 55, a(2) = 166265

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 54

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 783 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                  1 + 1566 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+783*a(n-1)^2)/(1+1566*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 56, a(2) = 175504

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 55

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 3245 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 3245 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+3245*a(n-1)^2)/(2+3245*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 57, a(2) = 185079

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 56

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 840 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                  1 + 1680 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+840*a(n-1)^2)/(1+1680*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 58, a(2) = 194996

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 57

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 3477 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 3477 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+3477*a(n-1)^2)/(2+3477*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 59, a(2) = 205261

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 58

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 899 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                  1 + 1798 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+899*a(n-1)^2)/(1+1798*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 60, a(2) = 215880

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 59

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 3717 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 3717 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+3717*a(n-1)^2)/(2+3717*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 61, a(2) = 226859

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 60

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            2 a(n - 1) (1 + 960 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                  1 + 1920 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+960*a(n-1)^2)/(1+1920*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 62, a(2) = 238204

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 61

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 3965 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 3965 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+3965*a(n-1)^2)/(2+3965*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 63, a(2) = 249921

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 62

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 1023 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 2046 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+1023*a(n-1)^2)/(1+2046*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 64, a(2) = 262016

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 63

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 4221 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 4221 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+4221*a(n-1)^2)/(2+4221*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 65, a(2) = 274495

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 64

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 1088 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 2176 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+1088*a(n-1)^2)/(1+2176*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 66, a(2) = 287364

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 65

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 4485 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 4485 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+4485*a(n-1)^2)/(2+4485*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 67, a(2) = 300629

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 66

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 1155 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 2310 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+1155*a(n-1)^2)/(1+2310*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 68, a(2) = 314296

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 67

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 4757 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 4757 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+4757*a(n-1)^2)/(2+4757*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 69, a(2) = 328371

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 68

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 1224 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 2448 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+1224*a(n-1)^2)/(1+2448*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 70, a(2) = 342860

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 69

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 5037 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 5037 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+5037*a(n-1)^2)/(2+5037*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 71, a(2) = 357769

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 70

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 1295 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 2590 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+1295*a(n-1)^2)/(1+2590*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 72, a(2) = 373104

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 71

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 5325 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 5325 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+5325*a(n-1)^2)/(2+5325*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 73, a(2) = 388871

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 72

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 1368 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 2736 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+1368*a(n-1)^2)/(1+2736*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 74, a(2) = 405076

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 73

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 5621 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 5621 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+5621*a(n-1)^2)/(2+5621*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 75, a(2) = 421725

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 74

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 1443 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 2886 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+1443*a(n-1)^2)/(1+2886*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 76, a(2) = 438824

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 75

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 5925 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 5925 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+5925*a(n-1)^2)/(2+5925*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 77, a(2) = 456379

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 76

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 1520 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 3040 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+1520*a(n-1)^2)/(1+3040*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 78, a(2) = 474396

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 77

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 6237 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 6237 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+6237*a(n-1)^2)/(2+6237*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 79, a(2) = 492881

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 78

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 1599 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 3198 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+1599*a(n-1)^2)/(1+3198*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 80, a(2) = 511840

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 79

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 6557 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 6557 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+6557*a(n-1)^2)/(2+6557*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 81, a(2) = 531279

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 80

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 1680 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 3360 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+1680*a(n-1)^2)/(1+3360*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 82, a(2) = 551204

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 81

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 6885 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 6885 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+6885*a(n-1)^2)/(2+6885*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 83, a(2) = 571621

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 82

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 1763 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 3526 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+1763*a(n-1)^2)/(1+3526*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 84, a(2) = 592536

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 83

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 7221 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 7221 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+7221*a(n-1)^2)/(2+7221*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 85, a(2) = 613955

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 84

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 1848 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 3696 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+1848*a(n-1)^2)/(1+3696*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 86, a(2) = 635884

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 85

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 7565 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 7565 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+7565*a(n-1)^2)/(2+7565*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 87, a(2) = 658329

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 86

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 1935 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 3870 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+1935*a(n-1)^2)/(1+3870*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 88, a(2) = 681296

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 87

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 7917 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 7917 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+7917*a(n-1)^2)/(2+7917*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 89, a(2) = 704791

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 88

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 2024 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 4048 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+2024*a(n-1)^2)/(1+4048*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 90, a(2) = 728820

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 89

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 8277 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 8277 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+8277*a(n-1)^2)/(2+8277*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 91, a(2) = 753389

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 90

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 2115 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 4230 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+2115*a(n-1)^2)/(1+4230*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 92, a(2) = 778504

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 91

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 8645 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 8645 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+8645*a(n-1)^2)/(2+8645*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 93, a(2) = 804171

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 92

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 2208 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 4416 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+2208*a(n-1)^2)/(1+4416*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 94, a(2) = 830396

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 93

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 9021 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 9021 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+9021*a(n-1)^2)/(2+9021*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 95, a(2) = 857185

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 94

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 2303 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 4606 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+2303*a(n-1)^2)/(1+4606*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 96, a(2) = 884544

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 95

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 9405 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 9405 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+9405*a(n-1)^2)/(2+9405*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 97, a(2) = 912479

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 96

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 2400 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 4800 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+2400*a(n-1)^2)/(1+4800*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 98, a(2) = 940996

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 97

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                             a(n - 1) (4 + 9797 a(n - 1) )
                      a(n) = -----------------------------
                                                   2
                                  2 + 9797 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+9797*a(n-1)^2)/(2+9797*a(n-2)^2)
                       subject to the initial conditions

                            a(1) = 99, a(2) = 970101

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 98

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 2499 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 4998 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+2499*a(n-1)^2)/(1+4998*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 100, a(2) = 999800

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                                Prop. Number, 99

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 10197 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 10197 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+10197*a(n-1)^2)/(2+10197*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 101, a(2) = 1030099

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 100

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 2600 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 5200 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+2600*a(n-1)^2)/(1+5200*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 102, a(2) = 1061004

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 101

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 10605 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 10605 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+10605*a(n-1)^2)/(2+10605*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 103, a(2) = 1092521

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 102

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 2703 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 5406 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+2703*a(n-1)^2)/(1+5406*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 104, a(2) = 1124656

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 103

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 11021 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 11021 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+11021*a(n-1)^2)/(2+11021*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 105, a(2) = 1157415

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 104

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 2808 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 5616 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+2808*a(n-1)^2)/(1+5616*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 106, a(2) = 1190804

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 105

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 11445 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 11445 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+11445*a(n-1)^2)/(2+11445*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 107, a(2) = 1224829

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 106

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 2915 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 5830 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+2915*a(n-1)^2)/(1+5830*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 108, a(2) = 1259496

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 107

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 11877 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 11877 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+11877*a(n-1)^2)/(2+11877*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 109, a(2) = 1294811

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 108

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 3024 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 6048 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+3024*a(n-1)^2)/(1+6048*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 110, a(2) = 1330780

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 109

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 12317 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 12317 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+12317*a(n-1)^2)/(2+12317*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 111, a(2) = 1367409

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 110

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 3135 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 6270 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+3135*a(n-1)^2)/(1+6270*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 112, a(2) = 1404704

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 111

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 12765 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 12765 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+12765*a(n-1)^2)/(2+12765*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 113, a(2) = 1442671

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 112

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 3248 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 6496 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+3248*a(n-1)^2)/(1+6496*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 114, a(2) = 1481316

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 113

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 13221 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 13221 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+13221*a(n-1)^2)/(2+13221*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 115, a(2) = 1520645

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 114

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 3363 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 6726 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+3363*a(n-1)^2)/(1+6726*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 116, a(2) = 1560664

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 115

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 13685 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 13685 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+13685*a(n-1)^2)/(2+13685*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 117, a(2) = 1601379

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 116

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 3480 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 6960 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+3480*a(n-1)^2)/(1+6960*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 118, a(2) = 1642796

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 117

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 14157 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 14157 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+14157*a(n-1)^2)/(2+14157*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 119, a(2) = 1684921

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 118

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 3599 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 7198 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+3599*a(n-1)^2)/(1+7198*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 120, a(2) = 1727760

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 119

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 14637 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 14637 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+14637*a(n-1)^2)/(2+14637*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 121, a(2) = 1771319

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 120

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 3720 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 7440 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+3720*a(n-1)^2)/(1+7440*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 122, a(2) = 1815604

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 121

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 15125 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 15125 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+15125*a(n-1)^2)/(2+15125*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 123, a(2) = 1860621

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 122

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 3843 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 7686 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+3843*a(n-1)^2)/(1+7686*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 124, a(2) = 1906376

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 123

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 15621 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 15621 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+15621*a(n-1)^2)/(2+15621*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 125, a(2) = 1952875

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 124

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 3968 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 7936 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+3968*a(n-1)^2)/(1+7936*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 126, a(2) = 2000124

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 125

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 16125 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 16125 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+16125*a(n-1)^2)/(2+16125*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 127, a(2) = 2048129

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 126

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 4095 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 8190 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+4095*a(n-1)^2)/(1+8190*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 128, a(2) = 2096896

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 127

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 16637 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 16637 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+16637*a(n-1)^2)/(2+16637*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 129, a(2) = 2146431

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 128

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 4224 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 8448 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+4224*a(n-1)^2)/(1+8448*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 130, a(2) = 2196740

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 129

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 17157 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 17157 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+17157*a(n-1)^2)/(2+17157*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 131, a(2) = 2247829

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 130

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 4355 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 8710 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+4355*a(n-1)^2)/(1+8710*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 132, a(2) = 2299704

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 131

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 17685 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 17685 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+17685*a(n-1)^2)/(2+17685*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 133, a(2) = 2352371

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 132

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 4488 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 8976 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+4488*a(n-1)^2)/(1+8976*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 134, a(2) = 2405836

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 133

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 18221 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 18221 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+18221*a(n-1)^2)/(2+18221*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 135, a(2) = 2460105

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 134

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 4623 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 9246 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+4623*a(n-1)^2)/(1+9246*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 136, a(2) = 2515184

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 135

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 18765 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 18765 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+18765*a(n-1)^2)/(2+18765*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 137, a(2) = 2571079

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 136

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 4760 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 9520 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+4760*a(n-1)^2)/(1+9520*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 138, a(2) = 2627796

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 137

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 19317 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 19317 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+19317*a(n-1)^2)/(2+19317*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 139, a(2) = 2685341

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 138

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 4899 a(n - 1) )
                     a(n) = -------------------------------
                                                   2
                                  1 + 9798 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+4899*a(n-1)^2)/(1+9798*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 140, a(2) = 2743720

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 139

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 19877 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 19877 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+19877*a(n-1)^2)/(2+19877*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 141, a(2) = 2802939

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 140

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 5040 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 10080 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+5040*a(n-1)^2)/(1+10080*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 142, a(2) = 2863004

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 141

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 20445 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 20445 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+20445*a(n-1)^2)/(2+20445*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 143, a(2) = 2923921

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 142

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 5183 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 10366 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+5183*a(n-1)^2)/(1+10366*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 144, a(2) = 2985696

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 143

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 21021 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 21021 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+21021*a(n-1)^2)/(2+21021*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 145, a(2) = 3048335

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 144

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 5328 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 10656 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+5328*a(n-1)^2)/(1+10656*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 146, a(2) = 3111844

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 145

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 21605 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 21605 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+21605*a(n-1)^2)/(2+21605*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 147, a(2) = 3176229

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 146

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 5475 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 10950 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+5475*a(n-1)^2)/(1+10950*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 148, a(2) = 3241496

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 147

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 22197 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 22197 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+22197*a(n-1)^2)/(2+22197*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 149, a(2) = 3307651

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 148

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 5624 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 11248 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+5624*a(n-1)^2)/(1+11248*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 150, a(2) = 3374700

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 149

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 22797 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 22797 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+22797*a(n-1)^2)/(2+22797*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 151, a(2) = 3442649

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 150

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 5775 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 11550 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+5775*a(n-1)^2)/(1+11550*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 152, a(2) = 3511504

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 151

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 23405 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 23405 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+23405*a(n-1)^2)/(2+23405*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 153, a(2) = 3581271

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 152

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 5928 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 11856 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+5928*a(n-1)^2)/(1+11856*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 154, a(2) = 3651956

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 153

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 24021 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 24021 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+24021*a(n-1)^2)/(2+24021*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 155, a(2) = 3723565

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 154

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 6083 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 12166 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+6083*a(n-1)^2)/(1+12166*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 156, a(2) = 3796104

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 155

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 24645 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 24645 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+24645*a(n-1)^2)/(2+24645*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 157, a(2) = 3869579

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 156

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 6240 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 12480 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+6240*a(n-1)^2)/(1+12480*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 158, a(2) = 3943996

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 157

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 25277 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 25277 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+25277*a(n-1)^2)/(2+25277*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 159, a(2) = 4019361

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 158

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 6399 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 12798 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+6399*a(n-1)^2)/(1+12798*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 160, a(2) = 4095680

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 159

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 25917 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 25917 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+25917*a(n-1)^2)/(2+25917*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 161, a(2) = 4172959

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 160

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 6560 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 13120 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+6560*a(n-1)^2)/(1+13120*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 162, a(2) = 4251204

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 161

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 26565 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 26565 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+26565*a(n-1)^2)/(2+26565*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 163, a(2) = 4330421

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 162

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 6723 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 13446 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+6723*a(n-1)^2)/(1+13446*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 164, a(2) = 4410616

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 163

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 27221 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 27221 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+27221*a(n-1)^2)/(2+27221*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 165, a(2) = 4491795

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 164

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 6888 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 13776 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+6888*a(n-1)^2)/(1+13776*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 166, a(2) = 4573964

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 165

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 27885 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 27885 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+27885*a(n-1)^2)/(2+27885*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 167, a(2) = 4657129

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 166

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 7055 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 14110 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+7055*a(n-1)^2)/(1+14110*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 168, a(2) = 4741296

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 167

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 28557 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 28557 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+28557*a(n-1)^2)/(2+28557*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 169, a(2) = 4826471

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 168

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 7224 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 14448 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+7224*a(n-1)^2)/(1+14448*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 170, a(2) = 4912660

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 169

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 29237 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 29237 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+29237*a(n-1)^2)/(2+29237*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 171, a(2) = 4999869

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 170

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 7395 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 14790 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+7395*a(n-1)^2)/(1+14790*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 172, a(2) = 5088104

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 171

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 29925 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 29925 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+29925*a(n-1)^2)/(2+29925*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 173, a(2) = 5177371

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 172

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 7568 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 15136 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+7568*a(n-1)^2)/(1+15136*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 174, a(2) = 5267676

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 173

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 30621 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 30621 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+30621*a(n-1)^2)/(2+30621*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 175, a(2) = 5359025

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 174

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 7743 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 15486 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+7743*a(n-1)^2)/(1+15486*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 176, a(2) = 5451424

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 175

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 31325 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 31325 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+31325*a(n-1)^2)/(2+31325*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 177, a(2) = 5544879

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 176

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 7920 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 15840 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+7920*a(n-1)^2)/(1+15840*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 178, a(2) = 5639396

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 177

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 32037 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 32037 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+32037*a(n-1)^2)/(2+32037*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 179, a(2) = 5734981

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 178

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 8099 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 16198 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+8099*a(n-1)^2)/(1+16198*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 180, a(2) = 5831640

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 179

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 32757 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 32757 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+32757*a(n-1)^2)/(2+32757*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 181, a(2) = 5929379

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 180

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 8280 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 16560 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+8280*a(n-1)^2)/(1+16560*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 182, a(2) = 6028204

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 181

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 33485 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 33485 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+33485*a(n-1)^2)/(2+33485*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 183, a(2) = 6128121

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 182

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 8463 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 16926 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+8463*a(n-1)^2)/(1+16926*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 184, a(2) = 6229136

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 183

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 34221 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 34221 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+34221*a(n-1)^2)/(2+34221*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 185, a(2) = 6331255

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 184

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 8648 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 17296 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+8648*a(n-1)^2)/(1+17296*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 186, a(2) = 6434484

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 185

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 34965 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 34965 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+34965*a(n-1)^2)/(2+34965*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 187, a(2) = 6538829

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 186

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 8835 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 17670 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+8835*a(n-1)^2)/(1+17670*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 188, a(2) = 6644296

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 187

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 35717 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 35717 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+35717*a(n-1)^2)/(2+35717*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 189, a(2) = 6750891

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 188

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 9024 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 18048 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+9024*a(n-1)^2)/(1+18048*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 190, a(2) = 6858620

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 189

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 36477 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 36477 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+36477*a(n-1)^2)/(2+36477*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 191, a(2) = 6967489

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 190

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 9215 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 18430 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+9215*a(n-1)^2)/(1+18430*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 192, a(2) = 7077504

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 191

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 37245 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 37245 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+37245*a(n-1)^2)/(2+37245*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 193, a(2) = 7188671

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 192

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 9408 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 18816 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+9408*a(n-1)^2)/(1+18816*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 194, a(2) = 7300996

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 193

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 38021 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 38021 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+38021*a(n-1)^2)/(2+38021*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 195, a(2) = 7414485

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 194

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 9603 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 19206 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+9603*a(n-1)^2)/(1+19206*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 196, a(2) = 7529144

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 195

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 38805 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 38805 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+38805*a(n-1)^2)/(2+38805*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 197, a(2) = 7644979

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 196

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 9800 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 19600 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+9800*a(n-1)^2)/(1+19600*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 198, a(2) = 7761996

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 197

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                        2
                            a(n - 1) (4 + 39597 a(n - 1) )
                     a(n) = ------------------------------
                                                   2
                                 2 + 39597 a(n - 2)



                          and in Maple input notation

a(n) = a(n-1)*(4+39597*a(n-1)^2)/(2+39597*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 199, a(2) = 7880201

                then, SURPRISE, SURPRISE, a(n) are ALL integers



                               Prop. Number, 198

                Let a(n) be the sequence of numbers defined via

                           the non-linear recurrence

                                                         2
                            2 a(n - 1) (1 + 9999 a(n - 1) )
                     a(n) = -------------------------------
                                                    2
                                  1 + 19998 a(n - 2)



                          and in Maple input notation

a(n) = 2*a(n-1)*(1+9999*a(n-1)^2)/(1+19998*a(n-2)^2)
                       subject to the initial conditions

                           a(1) = 200, a(2) = 7999600

                then, SURPRISE, SURPRISE, a(n) are ALL integers



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