A192923 Coefficient of x in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.

0, 1, 2, 4, 9, 19, 42, 91, 200, 437, 959, 2101, 4609, 10106, 22168, 48620, 106649, 233928, 513126, 1125541, 2468901, 5415578, 11879209, 26057330, 57157443, 125376341, 275016369, 603255761

Offset: 0

Author

Clark Kimberling, Jul 12 2011

Keywords

• nonn

Comments

The titular polynomial is defined by p(n,x) = p(n-1,x) +(x^2)*p(n-2,x), with p(0,x)=1, p(1,x)=x. For discussions of polynomial reduction, see A192232, A192744, and A192872.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,-3,-1).

Crossrefs

Cf. A192232, A192744, A192922.

Programs

GAP

a:=[0,1,2,4];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]-3*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Feb 06 2019

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1-2*x^2)/(1-2*x-2*x^2+3*x^3+x^4) )); // G. C. Greubel, Feb 06 2019

Mathematica

(See A192922.)
CoefficientList[Series[x*(1-2*x^2)/(1-2*x-2*x^2+3*x^3+x^4), {x, 0, 30}], x] (* G. C. Greubel, Jun 26 2017 *)

PARI

x='x+O('x^30); concat([0], Vec(x*(1-2*x^2)/(1-2*x-2*x^2+3*x^3+x^4) )) \\ G. C. Greubel, Jun 26 2017

Sage

(x*(1-2*x^2)/(1-2*x-2*x^2+3*x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 06 2019

Formula

a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) - a(n-4).

G.f.: x*(1-2*x^2) / ( 1-2*x-2*x^2+3*x^3+x^4 ). - R. J. Mathar, May 08 2014