A192922 Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.

1, 0, 1, 2, 5, 11, 25, 55, 122, 268, 590, 1295, 2844, 6240, 13693, 30039, 65900, 144559, 317108, 695595, 1525829, 3346965, 7341695, 16104238, 35325142, 77486710, 169969295, 372832346

Offset: 0

Author

Clark Kimberling, Jul 12 2011

Keywords

• nonn

Comments

The titular polynomial is defined by p(n,x) = p(n-1,x) +(x^2)*p(n-2,x), with p(0,x)=1, p(1,x)=x. For discussions of polynomial reduction, see A192232, A192744, and A192872.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,-3,-1).

Crossrefs

Cf. A192232, A192744, A192923.

Programs

GAP

a:=[1,0,1,2];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2] -3*a[n-3]-a[n-4]; od; a; # G. C. Greubel, Feb 06 2019

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x^2-2*x+3*x^3)/(1-2*x-2*x^2+3*x^3+x^4) )); // G. C. Greubel, Feb 06 2019

Mathematica

q = x^2; s = x + 1; z = 28;
p[0, x_] := 1; p[1, x_] := x;
p[n_, x_] := p[n - 1, x] + p[n - 2, x]*x^2;
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}]
  (* A192922 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]  (* A192923 *)
LinearRecurrence[{2,2,-3,-1}, {1,0,1,2}, 30] (* G. C. Greubel, Feb 06 2019 *)

PARI

my(x='x+O(x^30)); Vec((1-x^2-2*x+3*x^3)/(1-2*x-2*x^2+3*x^3+x^4)) \\ G. C. Greubel, Feb 06 2019

Sage

((1-x^2-2*x+3*x^3)/(1-2*x-2*x^2+3*x^3+x^4)).series(x, 30).coefficients(x, sparse=False)

Formula

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

G.f.: (1-x^2-2*x+3*x^3)/(1-2*x-2*x^2+3*x^3+x^4). - R. J. Mathar, May 08 2014