A295143 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 9, 25, 65, 162, 397, 966, 2340, 5658, 13669, 33010, 79704, 192434, 464589, 1121630, 2707868, 6537386, 15782661, 38102730, 91988144, 222079042

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, b(1) = 4
a(2) =2*a(1) + a(0) + b(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, A295141, A295142, A295144.

Programs

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = 2 a[ n - 1] + a[n - 2] + 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}]  (* A295143 *)
Table[b[n], {n, 0, 10}]

Formula

a(n+1)/a(n) -> 1 + sqrt(2).