A296227 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.

1, 2, 8, 34, 146, 628, 2703, 11632, 50057, 215415, 927016, 3989317, 17167612, 73879038, 317930779, 1368182139, 5887829959, 25337665679, 109038016813, 469233798454, 2019298993572, 8689843823858, 37395841786394, 160929127296116, 692541811472532

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..24.

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

Example

a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4
a(2) = a(0)*b(1) + a(1)*b(0) - 2 = 8
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...)

Mathematica

mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = - n + Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 200}]  (* A296227 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296228 *)

Crossrefs

Cf. A296000, A296228.

Sequence in context: A192402 A014445 A113440 * A034999 A067336 A245090

Adjacent sequences: A296224 A296225 A296226 * A296228 A296229 A296230

Keyword

nonn,easy

Author

Clark Kimberling, Dec 10 2017

Status

approved