A294385 Solution of the complementary equation a(n) = a(n-1)*b(n-2) + n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 8, 35, 179, 1079, 7559, 68038, 680388, 7484277, 89811334, 1167547353, 16345662954

Offset

0,2

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294381) for a guide to related sequences.

Links

Table of n, a(n) for n=0..12.

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

Example

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

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1]*b[n - 2] + n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294385 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A293076, A293765, A294381.

Sequence in context: A216541 A347896 A194090 * A020099 A182392 A005635

Adjacent sequences: A294382 A294383 A294384 * A294386 A294387 A294388

Keyword

nonn,more

Author

Clark Kimberling, Oct 29 2017

Status

approved