A022319 a(n) = a(n-1) + a(n-2) + 1 for n > 1, a(0)=1, a(1)=5.

1, 5, 7, 13, 21, 35, 57, 93, 151, 245, 397, 643, 1041, 1685, 2727, 4413, 7141, 11555, 18697, 30253, 48951, 79205, 128157, 207363, 335521, 542885, 878407, 1421293, 2299701, 3720995, 6020697, 9741693

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• Diana Savin and Elif Tan, On Companion sequences associated with Leonardo quaternions: Applications over finite fields, arXiv:2403.01592 [math.CO], 2024. See p. 2.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Crossrefs

Cf. A192762 (partial sums).

Programs

Haskell

a022319 n = a022319_list !! (n-1)
a022319_list = 1 : 5 : zipWith (+)
   (map (+ 1) a022319_list) (tail a022319_list)
-- Reinhard Zumkeller, May 26 2013

Maple

with(combinat): seq(fibonacci(n-2)+fibonacci(n+4)-1, n=0..29); # Zerinvary Lajos, Feb 01 2008

Mathematica

LinearRecurrence[{2, 0, -1}, {1, 5, 7}, 40] (* Harvey P. Dale, Nov 19 2014 *)

PARI

x='x+O('x^50); Vec((1 +3*x -3*x^2)/((1-x)*(1 -x -x^2))) \\ G. C. Greubel, Aug 25 2017

Formula

a(n) = Fibonacci(n-2) + Fibonacci(n+4) - 1. - Zerinvary Lajos, Feb 01 2008

From R. J. Mathar, Apr 07 2011: (Start)

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

a(n) = A022112(n) - 1. (End)