A192875 Coefficient of x in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

0, 1, 3, 11, 37, 119, 391, 1257, 4087, 13195, 42757, 138271, 447615, 1448249, 4687071, 15166963, 49082501, 158832391, 513995543, 1663319433, 5382623015, 17418520571, 56367538373, 182409150671, 590288468367, 1910213517529

Offset: 0

Author

Clark Kimberling, Jul 11 2011

Keywords

• nonn

Comments

The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x, and p(n,x) = x*p(n-1,x) + 2*(x^2)*p(n-1,x) + 1. See A192872.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,6,-5,-6,4).

Crossrefs

Cf. A192872, A192874.

Programs

GAP

a:=[0,1,3,11,37];; for n in [6..30] do a[n]:=2*a[n-1]+6*a[n-2] - 5*a[n-3]-6*a[n-4]+4*a[n-5]; od; a; # G. C. Greubel, Jan 08 2019

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1+2*x)*(1-x+x^2)/((1-x)*(1+ x-x^2)*(1-2*x-4*x^2)) )); // G. C. Greubel, Jan 08 2019

Maple

seq(coeff(series(x*(1+2*x)*(1-x+x^2)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2)),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Jan 08 2019

Mathematica

(See A192874.)
LinearRecurrence[{2,6,-5,-6,4}, {0,1,3,11,37}, 30] (* G. C. Greubel, Jan 08 2019 *)

PARI

my(x='x+O('x^30)); concat([0], Vec(x*(1+2*x)*(1-x+x^2)/((1-x)*(1+ x-x^2)*(1-2*x-4*x^2)))) \\ G. C. Greubel, Jan 08 2019

Sage

(x*(1+2*x)*(1-x+x^2)/((1-x)*(1+ x-x^2)*(1-2*x-4*x^2))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jan 08 2019

Formula

a(n) = 2*a(n-1) + 6*a(n-2) - 5*a(n-3) - 6*a(n-4) + 4*a(n-5).

G.f.: x*(1+2*x)*(1-x+x^2) / ( (1-x)*(1+x-x^2)*(1-2*x-4*x^2)). - R. J. Mathar, May 06 2014