A295057 Solution of the complementary equation a(n) = 2*a(n-1) + b(n-1), where a(0) = 2, a(1) = 5, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.

2, 5, 13, 30, 66, 139, 286, 581, 1172, 2355, 4722, 9458, 18931, 37878, 75773, 151564, 303147, 606314, 1212649, 2425320, 4850663, 9701350, 19402725, 38805476, 77610979, 155221986, 310444001, 620888033

Offset

0,1

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..27.

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

Example

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

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 2; a[1] = 5; b[0] = 1;
a[n_] := a[n] = 2 a[n - 1] + 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}]  (* A295057 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053.

Sequence in context: A282153 A054127 A184052 * A309535 A018012 A359673

Adjacent sequences: A295054 A295055 A295056 * A295058 A295059 A295060

Keyword

nonn,easy

Author

Clark Kimberling, Nov 18 2017

Status

approved