OFFSET
0,3
COMMENTS
The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x) + 1, with p(0,x) = 1, p(1,x) = x + 1.
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).
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
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 *)
PROG
(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
(Magma) m:=30; R<x>:=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
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 12 2011
STATUS
approved