A295135 Solution of the complementary equation a(n) = 3*a(n-1) + b(n-1) - 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 8, 27, 85, 260, 787, 2369, 7116, 21358, 64085, 192267, 576814, 1730456, 5191383, 15574165, 46722512, 140167554

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences.

Links

Table of n, a(n) for n=0..17.

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

Example

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

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 3 a[n - 1] + b[n - 1] -2;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A295135 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053.

Sequence in context: A102759 A292698 A261056 * A076884 A138388 A138386

Adjacent sequences: A295132 A295133 A295134 * A295136 A295137 A295138

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved