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A192922
Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.
2
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.
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 12 2011
STATUS
approved