A293767 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 7, 14, 26, 47, 81, 137, 228, 376, 616, 1006, 1637, 2659, 4313, 6990, 11322, 18332, 29675, 48029, 77727, 125780, 203533, 329340, 532901, 862270, 1395201, 2257502, 3652735, 5910270, 9563039, 15473344, 25036419, 40509800, 65546257, 106056096

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) - 1 = 7;
Complement: (b(n)) = (2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, ...)
		

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] - 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}]  (* A293767 *)
Table[b[n], {n, 0, 10}]