A192907 Coefficient of x in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

0, 1, 4, 12, 37, 116, 364, 1141, 3576, 11208, 35129, 110104, 345096, 1081625, 3390108, 10625524, 33303293, 104381612, 327160468, 1025410221, 3213915568, 10073288784, 31572437041, 98956636912

Offset: 0

Author

Clark Kimberling, Jul 12 2011

Keywords

• nonn

Comments

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

Links

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

Crossrefs

Cf. A192906.

Programs

GAP

a:=[0,1,4,12];; for n in [5..30] do a[n]:=3*a[n-1]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jan 11 2019

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1+x)/(1-3*x-x^3-x^4) )); // G. C. Greubel, Jan 11 2019

Mathematica

(See A192906.)
LinearRecurrence[{3,0,1,1}, {0,1,4,12}, 30] (* G. C. Greubel, Jan 11 2019 *)

PARI

my(x='x+O('x^30)); concat([0], Vec(x*(1+x)/(1-3*x-x^3-x^4))) \\ G. C. Greubel, Jan 11 2019

Sage

(x*(1+x)/(1-3*x-x^3-x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 11 2019

Formula

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

G.f. x*(1+x)/( 1-3*x-x^3-x^4 ). - R. J. Mathar, Jul 13 2011