A294557 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n - 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 11, 24, 49, 90, 159, 272, 457, 759, 1250, 2046, 3336, 5425, 8807, 14281, 23140, 37476, 60674, 98211, 158949, 257228, 416249, 673552, 1089879, 1763512, 2853475, 4617074, 7470639, 12087806, 19558541, 31646446, 51205089, 82851640, 134056837, 216908588

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

Links

Table of n, a(n) for n=0..35.

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

Example

a(0) = 1, a(1) = 2, b(0) = 3, so that
b(1) = 4 (least "new number")
a(2)  = a(1) + a(0) + b(1) + b(0) + 1 = 11
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 12, 13, 14, 15, 16, ...)

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] + 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, 40}]  (* A294557 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622, A294532.

Sequence in context: A297545 A256905 A294547 * A009189 A370338 A012213

Adjacent sequences: A294554 A294555 A294556 * A294558 A294559 A294560

Keyword

nonn,easy

Author

Clark Kimberling, Nov 15 2017

Extensions

Definition corrected by Georg Fischer, Sep 27 2020

Status

approved