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

0,3

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).

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

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 *)

Prog

(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
(Magma) m:=30; R<x>:=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
(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
(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

Crossrefs

Cf. A192872, A192874.

Sequence in context: A007615 A065540 A084171 * A118044 A027062 A134757

Adjacent sequences: A192872 A192873 A192874 * A192876 A192877 A192878

Keyword

nonn

Author

Clark Kimberling, Jul 11 2011

Status

approved