A293058 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 3, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 9, 19, 36, 64, 110, 185, 308, 507, 830, 1353, 2200, 3571, 5790, 9381, 15192, 24596, 39812, 64433, 104271, 168731, 273030, 441790, 714850, 1156671, 1871553, 3028257, 4899844, 7928136, 12828016, 20756189, 33584243, 54340472, 87924756, 142265270

Offset: 0

Author

Clark Kimberling, Oct 28 2017

Keywords

• nonn
• easy

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A293076 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) + b(0) + 3 = 9;
a(3) = a(2) + a(1) + b(1) + 1 = 19.
Complement: (b(n)) = (2,4,5,6,7,8,10,11,12,13,14,...)
		

Crossrefs

Cf. A001622 (golden ratio), A293076.

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 - 2] + 3;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A293316 *)
Table[b[n], {n, 0, 10}]