A192876 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

1, 1, 4, 9, 31, 94, 309, 989, 3212, 10373, 33595, 108670, 351729, 1138113, 3683172, 11918737, 38570247, 124815294, 403911805, 1307084405, 4229816636, 13687969901, 44295207939, 143342292894, 463865421721, 1501100008249

Offset: 0

Author

Clark Kimberling, Jul 11 2011

Keywords

• nonn

Comments

The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x + 1, 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).

Crossrefs

Cf. A192872, A192877.

Programs

GAP

a:=[1,1,4,9,31];; 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

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x-4*x^2)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2)) )); // G. C. Greubel, Jan 08 2019

Maple

seq(coeff(series((1-x-4*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

q = x^2; s = x + 1; z = 26;
p[0, x_] := 1; p[1, x_] := x + 1;
p[n_, x_] := p[n - 1, x]*x + 2*p[n - 2, x]*x^2 + 1;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] := FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]  (* A192876 *)
u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]  (* A192877 *)
FindLinearRecurrence[u0]
FindLinearRecurrence[u1]
LinearRecurrence[{2,6,-5,-6,4},{1,1,4,9,31},26] (* Ray Chandler, Aug 02 2015 *)

PARI

my(x='x+O('x^30)); Vec((1-x-4*x^2)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2) )) \\ G. C. Greubel, Jan 08 2019

Sage

((1-x-4*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

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.: (1-x-4*x^2) / ( (1-x)*(1+x-x^2)*(1-2*x-4*x^2) ). - R. J. Mathar, May 06 2014