A294365 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 10, 21, 41, 74, 129, 219, 367, 607, 997, 1629, 2653, 4311, 6995, 11339, 18369, 29745, 48154, 77941, 126139, 204126, 330313, 534489, 864854, 1399397, 2264307, 3663762, 5928129, 9591953, 15520146, 25112165, 40632379, 65744614, 106377065, 172121753

Offset: 0

Author

Clark Kimberling, Oct 29 2017

Keywords

• nonn
• easy

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:

Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio. See A293358 for a guide to related sequences.

Examples

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2)  = a(1) + a(0) + b(1) + 2 = 10;
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, ...)
		

Links

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

Crossrefs

Cf. A001622 (golden ratio), A293765.

Programs

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294365 *)
Table[b[n], {n, 0, 10}]