A296223 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 9, 34, 124, 453, 1654, 6040, 22055, 80532, 294058, 1073735, 3920679, 14316124, 52274468, 190877084, 696976221, 2544966858, 9292793804, 33932079081, 123900951107, 452416889887, 1651973131976, 6032080786047, 22025781112962, 80425818360771

Offset

0,2

Comments

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.

Links

Table of n, a(n) for n=0..25.

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Example

a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
a(2) = a(0)*b(1) + a(1)*b(0) - 1 = 9
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, ...)

Mathematica

mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}] - 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
u = Table[a[n], {n, 0, 200}]  (* A296223 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296224 *)

Crossrefs

Cf. A296000, A296224.

Sequence in context: A149006 A149007 A053791 * A045627 A007722 A149008

Adjacent sequences: A296220 A296221 A296222 * A296224 A296225 A296226

Keyword

nonn,easy

Author

Clark Kimberling, Dec 10 2017

Status

approved