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

0,4

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).

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

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 *)

Prog

(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
(Magma) m:=30; R<x>:=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
(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)
(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

Crossrefs

Cf. A192232, A192744, A192923.

Sequence in context: A092685 A172481 A151529 * A215091 A017919 A017920

Adjacent sequences: A192919 A192920 A192921 * A192923 A192924 A192925

Keyword

nonn

Author

Clark Kimberling, Jul 12 2011

Status

approved