A295136 Solution of the complementary equation a(n) = 3*a(n-1) + b(n-1) - 3, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 7, 23, 72, 221, 669, 2014, 6050, 18159, 54487, 163472, 490428, 1471297, 4413905, 13241730, 39725206, 119175635, 357526923

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..18.

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) - 3 = 7
Complement: (b(n)) = (3, 4, 5, 6, 8, 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] - 3;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A295136 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053.

Sequence in context: A085387 A147970 A027139 * A192906 A217664 A365692

Adjacent sequences: A295133 A295134 A295135 * A295137 A295138 A295139

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved