A192910 Coefficient of x in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

0, 1, 4, 13, 42, 133, 418, 1311, 4110, 12883, 40380, 126563, 396684, 1243317, 3896896, 12213937, 38281814, 119985657, 376067806, 1178699171, 3694364986, 11579148423, 36292212248, 113749700903, 356522616120, 1117439209033, 3502359540252

Offset

0,3

Comments

The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x) + 1, 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 (4,-3,1,0,-1).

Formula

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

G.f.: x*(1+x)*(1-x+x^2)/((1-x)*(1-3*x-x^3-x^4)). - R. J. Mathar, Jul 13 2011

Mathematica

(See A192909.)
LinearRecurrence[{4, -3, 1, 0, -1}, {0, 1, 4, 13, 42}, 30] (* G. C. Greubel, Jan 12 2019 *)

Prog

(PARI) my(x='x+O('x^30)); concat([0], Vec(x*(1+x)*(1-x+x^2)/((1-x)*(1-3*x -x^3-x^4)))) \\ G. C. Greubel, Jan 12 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1+x)*(1-x+x^2)/((1-x)*(1-3*x-x^3-x^4)) )); // G. C. Greubel, Jan 12 2019
(Sage) (x*(1+x)*(1-x+x^2)/((1-x)*(1-3*x-x^3-x^4))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 12 2019
(GAP) a:=[0, 1, 4, 13, 42];; for n in [6..30] do a[n]:=4*a[n-1]-3*a[n-2] + a[n-3]-a[n-5]; od; a; # G. C. Greubel, Jan 12 2019

Crossrefs

Cf. A192232, A192744, A192872, A192904, A192909.

Sequence in context: A287349 A000640 A199842 * A289807 A022029 A010919

Adjacent sequences: A192907 A192908 A192909 * A192911 A192912 A192913

Keyword

nonn

Author

Clark Kimberling, Jul 12 2011

Status

approved