A294556 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, 13, 28, 57, 104, 183, 312, 523, 866, 1423, 2327, 3792, 6164, 10004, 16219, 26277, 42553, 68890, 111506, 180462, 292037, 472571, 764683, 1237332, 2002097, 3239515, 5241701, 8481308, 13723104, 22204510, 35927715, 58132329, 94060151, 152192590, 246252854

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) + 3 = 13
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, ...)

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] + 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}]  (* A294556 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622, A294532.

Sequence in context: A294555 A031090 A358296 * A294559 A041017 A033837

Adjacent sequences: A294553 A294554 A294555 * A294557 A294558 A294559

Keyword

nonn,easy

Author

Clark Kimberling, Nov 15 2017

Extensions

Conjectured g.f. removed by Alois P. Heinz, Jun 25 2018

Definition corrected by Georg Fischer, Sep 27 2020

Status

approved