A295054 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(1) + b(2) + ... + 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, 7, 18, 40, 81, 153, 276, 482, 823, 1383, 2298, 3788, 6209, 10137, 16505, 26821, 43526, 70569, 114340, 185178, 299812, 485310, 785469, 1271154, 2057027, 3328615, 5386107, 8715219, 14101856, 22817639, 36920094, 59738368, 96659134

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..33.

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(1) + a(0) + b(1) = 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;
a[n_] := a[n] = a[n - 1] + a[n - 2] + Sum[b[k], {k, 1, 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}]  (* A295054 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053.

Sequence in context: A229183 A051743 A054111 * A192955 A055503 A077802

Adjacent sequences: A295051 A295052 A295053 * A295055 A295056 A295057

Keyword

nonn,easy

Author

Clark Kimberling, Nov 18 2017

Status

approved