A122100 a(n) = 3*a(n-1) - a(n-3) for n>2, with a(0)=1, a(1)=-1, a(2)=0.

1, -1, 0, -1, -2, -6, -17, -49, -141, -406, -1169, -3366, -9692, -27907, -80355, -231373, -666212, -1918281, -5523470, -15904198, -45794313, -131859469, -379674209, -1093228314, -3147825473, -9063802210, -26098178316, -75146709475, -216376326215, -623030800329, -1793945691512

Offset: 0

Author

Philippe Deléham, Oct 18 2006

Keywords

• sign
• easy
• less

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
• Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
• Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

Crossrefs

Cf. A052536, A122099.

Programs

GAP

a:=[1,-1,0];; for n in [4..40] do a[n]:=3*a[n-1]-a[n-3]; od; a; # G. C. Greubel, Oct 02 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-4*x+3*x^2)/(1-3*x+x^3) )); // G. C. Greubel, Oct 02 2019

Maple

seq(coeff(series((1-4*x+3*x^2)/(1-3*x+x^3), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Oct 02 2019

Mathematica

LinearRecurrence[{3,0,-1},{1,-1,0},40] (* Harvey P. Dale, Nov 14 2014 *)

PARI

Vec((1-4*x+3*x^2)/(1-3*x+x^3)+O(x^40)) \\ Charles R Greathouse IV, Jan 17 2012

Sage

def A122100_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-4*x+3*x^2)/(1-3*x+x^3)).list()
A122100_list(40) # G. C. Greubel, Oct 02 2019

Formula

G.f.: (1-4*x+3*x^2)/(1-3*x+x^3).