A296221 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, 11, 40, 146, 533, 1946, 7105, 25941, 94714, 345812, 1262601, 4609907, 16831321, 61453163, 224372837, 819212023, 2991040928, 10920647625, 39872588647, 145579582824, 531528442330, 1940673819263, 7085631873740, 25870488153041, 94456241758347

Offset: 0

Author

Clark Kimberling, Dec 09 2017

Keywords

• nonn
• easy

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.

Examples

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

Links

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

Crossrefs

Cf. A296000, A296222.

Programs

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}];  (* A296221 *)
Table[b[n], {n, 0, 20}]
t = N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
d = RealDigits[Last[t], 10][[1]] (* A296222 *)

Extensions

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