A022318 a(n) = a(n-1) + a(n-2) + 1, with a(0) = 1 and a(1) = 4.

1, 4, 6, 11, 18, 30, 49, 80, 130, 211, 342, 554, 897, 1452, 2350, 3803, 6154, 9958, 16113, 26072, 42186, 68259, 110446, 178706, 289153, 467860, 757014, 1224875, 1981890, 3206766, 5188657, 8395424

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• G. C. Greubel, 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. A122195.

Programs

Mathematica

LinearRecurrence[{2,0,-1}, {1,4,6}, 50] (* G. C. Greubel, Aug 25 2017 *)

PARI

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

Formula

From Ron Knott, Aug 25 2006: (Start)

a(n) = 2*A000045(n+2) + A000045(n) - 1.

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

a(0)=1, a(1)=4, a(2)=6, a(n) = 2*a(n-1) - a(n-3). (End)

a(n) - a(n-1) = A013655(n-1). - R. J. Mathar, May 06 2014