A192909 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.
1, 1, 3, 9, 27, 83, 259, 811, 2541, 7963, 24957, 78221, 245165, 768413, 2408415, 7548629, 23659463, 74155215, 232422687, 728476151, 2283243129, 7156307287, 22429820697, 70301181369, 220343094521, 690615411545, 2164577236699
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 (4,-3,1,0,-1).
Programs
-
GAP
a:=[1,1,3,9,27];; for n in [6..30] do a[n]:=4*a[n-1]-3*a[n-2] +a[n-3]-a[n-5]; od; a; # G. C. Greubel, Jan 11 2019
-
Magma
m:=30; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!( (x^2-x+1)*(x^2+2*x-1)/((1-x)*(x^4+x^3+3*x-1)) )); // G. C. Greubel, Jan 11 2019 -
Mathematica
u = 1; v = 1; a = 1; b = 1; c = 1; d = 1; e = 1; f = 1; q = x^2; s = u*x + v; z = 24; p[0, x_] := a; p[1, x_] := b*x + c p[n_, x_] := d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f; Table[Expand[p[n, x]], {n, 0, 8}] 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}]; u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192909 *) u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192910 *) Simplify[FindLinearRecurrence[u0]] Simplify[FindLinearRecurrence[u1]] LinearRecurrence[{4,-3,1,0,-1}, {1,1,3,9,27}, 30] (* G. C. Greubel, Jan 11 2019 *) -
PARI
my(x='x+O('x^30)); Vec((x^2-x+1)*(x^2+2*x-1)/((1-x)*(x^4+x^3+3*x -1))) \\ G. C. Greubel, Jan 11 2019 -
Sage
((x^2-x+1)*(x^2+2*x-1)/((1-x)*(x^4+x^3+3*x-1))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 11 2019
Formula
a(n) = 4*a(n-1) - 3*a(n-2) + a(n-3) - a(n-5).
G.f.: (x^2-x+1)*(x^2+2*x-1) / ( (1-x)*(x^4+x^3+3*x-1) ). - R. J. Mathar, Jul 13 2011
Comments