A295133 Solution of the complementary equation a(n) = 3*a(n-1) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 10, 35, 111, 340, 1028, 3093, 9290, 27882, 83659, 250991, 752988, 2258980, 6776957, 20330889, 60992686, 182978078, 548934255

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) = 10
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 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];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A295133 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053.

Sequence in context: A116898 A236377 A197556 * A100230 A220255 A146983

Adjacent sequences: A295130 A295131 A295132 * A295134 A295135 A295136

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved