A295055 Solution of the complementary equation a(n) = a(n-2) + b(0) + b(1) + ... + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 8, 14, 26, 39, 60, 83, 115, 150, 195, 245, 306, 373, 452, 538, 637, 744, 865, 995, 1140, 1295, 1467, 1650, 1851, 2064, 2296, 2541, 2806, 3085, 3385, 3700, 4037, 4390, 4767, 5161, 5580, 6017, 6480, 6962, 7471, 8000, 8557, 9135, 9742, 10371, 11030, 11712

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..47.

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

Example

a(0) = 1, a(1) = 2, b(0) = 3
b(1) = 4 (least "new number")
a(2) = a(0) + b(0) + b(1) = 8
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, ...)

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 2] + Sum[b[k], {k, 0, 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}]  (* A295055 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053.

Sequence in context: A120413 A161156 A125902 * A277276 A324785 A173974

Adjacent sequences: A295052 A295053 A295054 * A295056 A295057 A295058

Keyword

nonn,easy

Author

Clark Kimberling, Nov 18 2017

Status

approved