A289762 - OEIS

%I #45 Jul 17 2017 04:43:40

%S 1,1,4,2,2,4,9,5,3,3,5,9,16,10,6,4,4,6,10,16,25,17,11,7,5,5,7,11,17,

%T 25,36,26,18,12,8,6,6,8,12,18,26,36,49,37,27,19,13,9,7,7,9,13,19,27,

%U 37,49,64,50,38,28,20,14,10,8,8,10,14,20,28,38,50,64,81,65,51,39,29,21,15,11,9

%N Triangular array T(m,k) = (m+1-k)^2 + k - 1 with m (row) >= 1 and k (column) >= 1, read by rows.

%C The n-th row is of length = max(2n, 1) and the row sum is (2n^3 + 6n^2 - 2n) / 3.

%C Rows m = 2, 3, 5, 11, and 41 (Euler's lucky numbers) give the prime numbers generated by the famous polynomials, but twice each one between m^2.

%H Miquel Cerda, <a href="/A289762/b289762.txt">Table of n, a(n) for n = 1..306</a>

%H Miquel Cerda, <a href="/A289762/a289762.pdf">Triangle rows 1..41</a>

%H Miquel Cerda, <a href="/A289762/a289762_1.pdf">Isosceles triangle Rows 1..41</a>

%F The formula that gives the integers in the m-th rows can be expressed using quadratic polynomials:

%F for row m = 1, a(k) = k^2 - 3*k + 3

%F for row m = 2, a(k) = k^2 - 5*k + 8

%F for row m = 3, a(k) = k^2 - 7*k + 15

%F for row m = 4, a(k) = k^2 - 9*k + 24

%F for row m = 5, a(k) = k^2 - 11*k + 35

%F for row m = 6, a(k) = k^2 - 13*k + 48

%F etc.

%e The m-th row start and end: T(m,1) = m^2, ..., T(m,2m) = m^2.

%e In general T(m,k) = T(m,2m+1-k).

%e m\k    1     2     3     4     5     6     7     8     9     10

%e 1      1,    1,

%e 2      4,    2,    2,    4

%e 3      9,    5,    3,    3,    5,    9

%e 4      16,   10,   6,    4,    4,    6,    10,   16

%e 5      25,   17,   11,   7,    5,    5,    7,    11,   17,   25

%e 6      36,   26,   18,   12,   8,    6,    6,    8,    12,   18, ...

%e 7      49,   37,   27,   19,   13,   9,    7,    7,    9,    13, ...

%e 8      64,   50,   38,   28,   20,   14,   10,   8,    8,    10, ...

%e 9      81,   65,   51,   39,   29,   21,   15,   11,   9,    9, ...

%e 10     100,  82,   66,   52,   40,   30    22,   16,   12,   10, ...

%e The T(m,k) sequence as an isosceles triangle:

%e                                      1  1

%e                                  4   2  2  4

%e                              9   5   3  3  5  9

%e                          16  10  6   4  4  6  10  16

%e                      25  17  11  7   5  5  7  11  17  25

%e                  36  26  18  12  8   6  6  8  12  18  26  36

%e              49  37  27  19  13  9   7  7  9  13  19  27  37  49

%e          64  50  38  28  20  14  10  8  8  1  14  20  28  38  50  64

%e      81  65  51  39  29  21  15  11  9  9  11 15  21  29  39  51  65  81

%e 100  82  66  52  40  30  22  16  12  10 10 12 16  22  30  40  52  66  82  100

%t Table[(m + 1 - k)^2 + k - 1, {m, 0, 10}, {k, 2 m}] /. {} -> {0} // Flatten (* _Michael De Vlieger_, Jul 12 2017 *)

%o (PARI) T(m,k) = (m+1-k)^2+k-1 \\ _Charles R Greathouse IV_, Jul 12 2017

%Y m(41, k+1) = A060566(n), left and right border gives A000290(n).

%K nonn,tabf,easy

%O 1,3

%A _Miquel Cerda_, Jul 12 2017