A295142 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 9, 25, 64, 159, 389, 945, 2289, 5534, 13369, 32285, 77953, 188206, 454381, 1096985, 2648369, 6393742, 15435873, 37265509, 89966913, 217199358, 524365653, 1265930690

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..23.

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

Formula

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

Example

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

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = 2 a[ n - 1] + 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}]  (* A295142 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053, A295141, A295143, A295144.

Sequence in context: A375135 A292326 A195417 * A002064 A129589 A335472

Adjacent sequences: A295139 A295140 A295141 * A295143 A295144 A295145

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2017

Status

approved