A295134 Solution of the complementary equation a(n) = 3*a(n-1) + b(n-1) - 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 9, 31, 98, 300, 907, 2730, 8200, 24611, 73845, 221548, 664658, 1993989, 5981983, 17945966, 53837916, 161513767

Offset: 0

Author

Clark Kimberling, Nov 19 2017

Keywords

• nonn
• easy

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.

Examples

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

Links

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

Crossrefs

Cf. A295053.

Programs

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] -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}]  (* A295134 *)
Table[b[n], {n, 0, 10}]