A296001 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 10, 43, 185, 796, 3425, 14737, 63411, 272845, 1174000, 5051498, 21735632, 93524277, 402417118, 1731524071, 7450417675, 32057725596, 137938276110, 593522081260, 2553812262104, 10988566855385, 47281706383454, 203444160458068, 875381402033582

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> 4.302809183918588... (as in A296002). See A296000 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, so that
a(2) = a(0)*b(1) + a(1)*b(0) = 10
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, ...)

Mathematica

mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 2; b[0] = 3; a[n_] := a[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, 100}];  (* A296001 *)
t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]
Take[RealDigits[Last[t], 10][[1]], 100]  (* A296002 *)

Crossrefs

Cf. A296000, A296002.

Sequence in context: A286760 A197048 A175613 * A121949 A376225 A005144

Adjacent sequences: A295998 A295999 A296000 * A296002 A296003 A296004

Keyword

nonn,easy

Author

Clark Kimberling, Dec 04 2017

Extensions

Conjectured g.f. removed by Alois P. Heinz, Jun 25 2018

Status

approved