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

Author

Clark Kimberling, Nov 15 2017

Keywords

• nonn
• easy

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).

Examples

			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, ...)
		

Links

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

Crossrefs

Cf. A001622, A294532.

Programs

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}]

Extensions

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

Definition corrected by Georg Fischer, Sep 27 2020