A296227 - OEIS

%I #5 Dec 11 2017 14:31:00

%S 1,2,8,34,146,628,2703,11632,50057,215415,927016,3989317,17167612,

%T 73879038,317930779,1368182139,5887829959,25337665679,109038016813,

%U 469233798454,2019298993572,8689843823858,37395841786394,160929127296116,692541811472532

%N Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0) - n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%C The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.

%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.

%e a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4

%e a(2) = a(0)*b(1) + a(1)*b(0) - 2 = 8

%e Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...)

%t mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];

%t a[0] = 1; a[1] = 2; b[0] = 3;

%t a[n_] := a[n] = - n + Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];

%t b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

%t Table[a[n], {n, 0, 200}]  (* A296227 *)

%t Table[b[n], {n, 0, 20}]

%t N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];

%t RealDigits[Last[t], 10][[1]] (* A296228 *)

%Y Cf. A296000, A296228.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Dec 10 2017