A078007 Expansion of (1-x)/(1-x-2*x^2-x^3).

1, 0, 2, 3, 7, 15, 32, 69, 148, 318, 683, 1467, 3151, 6768, 14537, 31224, 67066, 144051, 309407, 664575, 1427440, 3065997, 6585452, 14144886, 30381787, 65257011, 140165471, 301061280, 646649233, 1388937264, 2983297010, 6407820771, 13763352055, 29562290607

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• nonn
• easy

Comments

Let X = the 3x3 matrix [0,1,0; 0,0,1; 1,2,1]. a(n) = center term of X^n; but A002478(n) = term (3,3) of X^n. - Gary W. Adamson, May 30 2008

First bisection of A058278. - Oboifeng Dira, Aug 04 2016

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,1).

Crossrefs

First differences of A002478.

Cf. A058278.

Programs

GAP

a:=[1,0,2];; for n in [4..40] do a[n]:=a[n-1]+2*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Jun 28 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x)/(1-x-2*x^2-x^3) )); // G. C. Greubel, Jun 28 2019

Mathematica

LinearRecurrence[{1,2,1}, {1,0,2}, 40] (* or *) CoefficientList[Series[(1 -x)/(1-x-2*x^2-x^3), {x,0,40}], x] (* G. C. Greubel, Jun 28 2019 *)

PARI

Vec((1-x)/(1-x-2*x^2-x^3)+O(x^40)) \\ Charles R Greathouse IV, Sep 26 2012

Sage

((1-x)/(1-x-2*x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 28 2019

Formula

a(n) = a(n-1) + 2*a(n-2) + a(n-3). - Ilya Gutkovskiy, Aug 06 2016