A192813 Coefficient of x in the reduction of the polynomial x^(2*n) + x^n + 1 by x^3 -> x + 1.

0, 1, 1, 3, 4, 6, 11, 18, 31, 53, 91, 158, 274, 477, 832, 1453, 2541, 4447, 7788, 13646, 23919, 41938, 73547, 129001, 226295, 397006, 696546, 1222153, 2144464, 3762921, 6603001, 11586843, 20332676, 35680278, 62613091, 109876418, 192817159

Offset

1,4

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1001

Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1,0,-1).

Formula

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

G.f.: -x^2*(x^4+2*x^3-x^2+x-1)/((x^3-x^2+2*x-1)*(x^3+x^2-1)). - Colin Barker, Nov 23 2012

Maple

seq(coeff(series((-x^2*(x^4+2*x^3-x^2+x-1))/((x^3-x^2+2*x-1)*(x^3+x^2-1)), x, n+1), x, n), n = 1 .. 40); # Muniru A Asiru, Jan 03 2019

Mathematica

(See A192812.)
LinearRecurrence[{2, 0, 0, -1, 0, -1}, {0, 1, 1, 3, 4, 6}, 40] (* G. C. Greubel, Jan 03 2019 *)

Prog

(PARI) my(x='x+O('x^40)); concat([0], Vec(x^2*(1-x+x^2-2*x^3-x^4)/((1-x^2-x^3)*(1-2*x+x^2-x^3)))) \\ G. C. Greubel, Jan 03 2019
(Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x^2*(1-x+x^2-2*x^3-x^4)/((1-x^2-x^3)*(1-2*x+x^2-x^3)) )); // G. C. Greubel, Jan 03 2019
(Sage) a=(x^2*(1-x+x^2-2*x^3-x^4)/((1-x^2-x^3)*(1-2*x+x^2-x^3)) ).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jan 03 2019

Crossrefs

Cf. A192744, A192232, A192812.

Sequence in context: A069825 A093040 A022935 * A048229 A002090 A199970

Adjacent sequences: A192810 A192811 A192812 * A192814 A192815 A192816

Keyword

nonn,easy

Author

Clark Kimberling, Jul 10 2011

Status

approved