A295147 Solution of the complementary equation a(n) = a(n-1) + 2*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, 8, 17, 39, 80, 167, 337, 682, 1368, 2745, 5495, 11000, 22006, 44024, 88055, 176123, 352254, 704522, 1409053, 2818121, 5636252, 11272520, 22545051, 45090119, 90180250, 180360518

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(1) = 8
Complement: (b(n)) = (3, 4, 5, 6, 7, 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 - 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}]  (* A295147 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053, A295145, A295146, A295148.

Sequence in context: A061150 A160189 A281470 * A215933 A328742 A341506

Adjacent sequences: A295144 A295145 A295146 * A295148 A295149 A295150

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved