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
Keywords
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).
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
Comments