A192874 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

1, 0, 4, 6, 26, 72, 246, 774, 2532, 8150, 26434, 85448, 276654, 895054, 2896788, 9373678, 30334682, 98163784, 317666758, 1027987894, 3326644036, 10765237670, 34837054674, 112735054856, 364818336766, 1180576879422

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, 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, A192875.

Programs

GAP

a:=[1,0,4,6,26];; 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

I:=[1,0,4,6,26]; [n le 5 select I[n] else 2*Self(n-1)+6*Self(n-2) -5*Self(n-3)-6*Self(n-4)+4*Self(n-5): n in [1..30]]; // G. C. Greubel, Jan 08 2019

Mathematica

q = x^2; s = x + 1; z = 26;
p[0, x_] := 1; p[1, x_] := x;
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}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192874 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192875 *)
LinearRecurrence[{2,6,-5,-6,4}, {1,0,4,6,26}, 30] (* G. C. Greubel, Jan 08 2019 *)

PARI

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

Sage

((x^2-x+1)*(4*x^2+x-1)/((x-1)*(x^2-x-1)*( 4*x^2+2*x-1))).series(x, 30).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.: (x^2-x+1)*(4*x^2+x-1) / ( (x-1)*(x^2-x-1)*(4*x^2+2*x-1) ). - R. J. Mathar, May 06 2014