A294421 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 10, 19, 36, 63, 108, 181, 302, 496, 812, 1323, 2151, 3491, 5660, 9170, 14852, 24044, 38919, 62987, 101931, 164944, 266902, 431874, 698805, 1130709, 1829545, 2960286, 4789864, 7750184, 12540083, 20290303, 32830425, 53120767, 85951232, 139072040

Offset: 0

Author

Clark Kimberling, Oct 31 2017

Keywords

• nonn
• easy

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294414 for a guide to related sequences.

Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.

Examples

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

Links

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

Crossrefs

Cf. A293076, A293765, A294414.

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] + 2 b[n - 1] - b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294421 *)
Table[b[n], {n, 0, 10}]