A295145 Solution of the complementary equation a(n) = a(n-1) + 2*a(n-2) + b(n-2), 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, 7, 15, 34, 70, 146, 295, 597, 1198, 2404, 4813, 9635, 19277, 38564, 77136, 154283, 308575, 617162, 1234334, 2468681, 4937373, 9874760, 19749532, 39499079, 78998171, 157996358

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..26.

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

Formula

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

Example

a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4
a(2) = a(1) + 2*a(0) + b(0) = 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; b[1] = 4;
a[n_] := a[n] = a[ n - 1] + 2 a[n - 2] + b[n - 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}]  (* A295145 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053, A295146, A295147, A295148.

Sequence in context: A345448 A209633 A216633 * A151998 A308631 A052130

Adjacent sequences: A295142 A295143 A295144 * A295146 A295147 A295148

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved