A296215 Solution of the complementary equation a(n) = a(1)*b(n-2) + a(2)*b(n-3) + ... + a(n-1)*b(0), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 6, 24, 87, 321, 1176, 4314, 15822, 58032, 212847, 780672, 2863317, 10501959, 38518662, 141277197, 518170812, 1900526031, 6970672818, 25566752964, 93772706622, 343935755925, 1261473710904, 4626782461218, 16969926331719, 62241612204120, 228287277978756

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..26.

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

Example

a(0) =1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5
a(2) = a(1)*b(0) = 6
Complement: (b(n)) = (2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, ...)

Mathematica

a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 1, n - 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}];  (* A296215 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296216 *)

Crossrefs

Cf. A296000, A296217.

Sequence in context: A132390 A363016 A327643 * A152761 A295761 A262349

Adjacent sequences: A296212 A296213 A296214 * A296216 A296217 A296218

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved