OFFSET
0,3
COMMENTS
The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x, and p(n,x) = x*p(n-1,x) + 2*(x^2)*p(n-1,x) + 1. See A192872.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,6,-5,-6,4).
FORMULA
a(n) = 2*a(n-1) + 6*a(n-2) - 5*a(n-3) - 6*a(n-4) + 4*a(n-5).
G.f.: (x^2-x+1)*(4*x^2+x-1) / ( (x-1)*(x^2-x-1)*(4*x^2+2*x-1) ). - R. J. Mathar, May 06 2014
MATHEMATICA
q = x^2; s = x + 1; z = 26;
p[0, x_] := 1; p[1, x_] := x;
p[n_, x_] := p[n - 1, x]*x + 2*p[n - 2, x]*x^2 + 1;
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}] (* A192874 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192875 *)
LinearRecurrence[{2, 6, -5, -6, 4}, {1, 0, 4, 6, 26}, 30] (* G. C. Greubel, Jan 08 2019 *)
PROG
(PARI) my(x='x+O('x^30)); Vec((x^2-x+1)*(4*x^2+x-1)/((x-1)*(x^2-x-1)*( 4*x^2+2*x-1))) \\ G. C. Greubel, Jan 08 2019
(Magma) I:=[1, 0, 4, 6, 26]; [n le 5 select I[n] else 2*Self(n-1)+6*Self(n-2) -5*Self(n-3)-6*Self(n-4)+4*Self(n-5): n in [1..30]]; // G. C. Greubel, Jan 08 2019
(Sage) ((x^2-x+1)*(4*x^2+x-1)/((x-1)*(x^2-x-1)*( 4*x^2+2*x-1))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 08 2019
(GAP) a:=[1, 0, 4, 6, 26];; for n in [6..30] do a[n]:=2*a[n-1]+6*a[n-2] - 5*a[n-3]-6*a[n-4]+4*a[n-5]; od; a; # G. C. Greubel, Jan 08 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 11 2011
STATUS
approved