A192810 Coefficient of x^2 in the reduction of the polynomial (x^2 + 2)^n by x^3 -> x^2 + 2.

0, 1, 5, 23, 109, 527, 2565, 12503, 60957, 297183, 1448821, 7063207, 34434061, 167870511, 818390501, 3989759863, 19450597117, 94824185471, 462280211797, 2253676033863, 10986963179245, 53562871542735, 261125950919109, 1273022903354903

Offset: 0

Author

Clark Kimberling, Jul 10 2011

Keywords

• nonn
• easy

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-12,8).

Crossrefs

Cf. A192744, A192232, A192808.

Programs

GAP

a:=[0,1,5];; for n in [4..25] do a[n]:=7*a[n-1]-12*a[n-2]+8*a[n-3]; od; Print(a); # Muniru A Asiru, Jan 02 2019

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1-2*x)/(1-7*x+12*x^2-8*x^3) )); // G. C. Greubel, Jan 02 2019

Mathematica

(See A192808.)
LinearRecurrence[{7,-12,8}, {0,1,5}, 30] (* G. C. Greubel, Jan 02 2019 *)
CoefficientList[Series[x (1-2x)/(1-7x+12x^2-8x^3),{x,0,30}],x] (* Harvey P. Dale, Aug 26 2021 *)

PARI

my(x='x+O('x^30)); concat([0], Vec(x*(1-2*x)/(1-7*x+12*x^2-8*x^3) )) \\ G. C. Greubel, Jan 02 2019

Sage

(x*(1-2*x)/(1-7*x+12*x^2-8*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 02 2019

Formula

a(n) = 7*a(n-1) - 12*a(n-2) + 8*a(n-3).

G.f.: x*(1-2*x)/(1-7*x+12*x^2-8*x^3). - Colin Barker, Jul 26 2012