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

0,2

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).

Links

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

Example

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.

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 *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A025560 A073618 A247651 * A221510 A109489 A169636

Adjacent sequences: A305743 A305744 A305745 * A305747 A305748 A305749

Keyword

nonn,easy

Author

Clark Kimberling, Jun 10 2018

Status

approved