A022410 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=3, a(1)=11.

3, 11, 15, 27, 43, 71, 115, 187, 303, 491, 795, 1287, 2083, 3371, 5455, 8827, 14283, 23111, 37395, 60507, 97903, 158411, 256315, 414727, 671043, 1085771, 1756815, 2842587, 4599403, 7441991, 12041395, 19483387, 31524783, 51008171, 82532955, 133541127

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1)

Crossrefs

Cf. A000032, A156279.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((3+5*x-7*x^2)/((x-1)*(x^2+x-1)))); // G. C. Greubel, Feb 28 2018

Mathematica

LinearRecurrence[{2, 0, -1}, {3, 11, 15}, 40] (* Bruno Berselli, Jul 27 2017 *)

PARI

Vec((3+5*x-7*x^2)/((x-1)*(x^2+x-1)) + O(x^50)) \\ Colin Barker, Jul 27 2017

Python

from sympy import lucas
def a(n): return 4 * lucas(n + 1) - 1
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 27 2017

Formula

From R. J. Mathar, Mar 11 2011: (Start)

a(n+1) - a(n) = A156279(n).

G.f.: (3 + 5*x - 7*x^2) / ((x - 1)*(x^2 + x - 1)).

(End)

a(n) = A156279(n+1) - 1. - Bruno Berselli, Jul 27 2017

From Colin Barker, Jul 27 2017: (Start)

a(n) = 2^(-n)*(-2^n + 2*(1-sqrt(5))^(1+n) + 2*(1+sqrt(5))^(1+n)).

a(n) = 2*a(n-1) - a(n-3) for n>2.

(End)