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

0,2

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

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 *)

Crossrefs

Cf. A296000, A296222.

Sequence in context: A108153 A010911 A052941 * A371872 A014301 A346194

Adjacent sequences: A296218 A296219 A296220 * A296222 A296223 A296224

Keyword

nonn,easy

Author

Clark Kimberling, Dec 09 2017

Extensions

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

Status

approved