A305746 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-3) + b(n), where a(0) = 1, a(1) = 2, a(2) = 3, b(0)= 4, b(1) = 5, b(2) = 6; b(3) = 7. See Comments.

1, 2, 3, 12, 30, 66, 130, 241, 429, 742, 1258, 2103, 3481, 5722, 9360, 15259, 24817, 40296, 65356, 105919, 171567, 277804, 449716, 727893, 1178011, 1906337, 3084813, 4991648, 8076993, 13069208, 21146804, 34216652, 55364134, 89581503, 144946394, 234528695

Offset: 0

Author

Clark Kimberling, Jun 10 2018

Keywords

• nonn
• easy

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values; a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio (A001622).

Examples

			a(0) = 1, a(1) = 2, a(2) = 3, b(0)= 4, b(1) = 5, b(2) = 6; b(3) = 7, and a(3) = 2*3 - 1 + 7 = 12.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A001622, A296245.

Programs

Mathematica

a[0] = 1; a[1] = 2; a[2] = 3; b[0] = 4; b[1] = 5; b[2] = 6; b[3] = 7;
a[n_] := a[n] = 2*a[n - 1] - a[n - 3] + b[n];
j = 1; While[j < 12, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, 60}]  (* A305746 *)