A088859 a(n) = L(n) + 2^n where L(n) = A000032(n) (the Lucas numbers).

3, 3, 7, 12, 23, 43, 82, 157, 303, 588, 1147, 2247, 4418, 8713, 17227, 34132, 67743, 134643, 267922, 533637, 1063703, 2121628, 4233907, 8452687, 16880898, 33722193, 67380307, 134656932, 269146103, 538020763, 1075602322, 2150493997

Offset: 0

Author

Miklos Kristof, Nov 25 2003

Keywords

• easy
• nonn

Comments

Lim_{n->infinity} a(n)/a(n-1) = 2.

Examples

			a(6) = 82 = L(6) + 2^6 = 18 + 64.
a(7) = 157 = 3*82 - 43 - 2*23 = 246 - 43 - 46.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (3, -1, -2).

Crossrefs

Cf. A000032.

Programs

Magma

[2^n+Lucas(n): n in [0..50]]; // Vincenzo Librandi, Apr 14 2011

Formula

G.f.: (3 - 6*x + 2*x^2) / (1 - 3*x + x^2 + 2*x^3)

a(n) = p^n + q^n + r^n, where p = (1+sqrt(5))/2, q = (1-sqrt(5))/2, and r = 2*p^n + q^n = L(n) = A000032(n), so a(n) = L(n) + 2^n

a(0)=3, a(1)=3, a(2)=7 and a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) for n >= 3.