A293350 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 2n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 10, 23, 46, 85, 150, 257, 432, 718, 1182, 1935, 3155, 5131, 8330, 13508, 21888, 35449, 57393, 92901, 150356, 243323, 393748, 637143, 1030966, 1668187, 2699234, 4367505, 7066826, 11434421, 18501340, 29935857, 48437296, 78373255, 126810656, 205184019

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) + 4 = 10;
a(3) = a(2) + a(1) + b(1) + 6 = 23.
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 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] + 2n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A293350 *)
Table[b[n], {n, 0, 10}]