OFFSET
1,5
LINKS
Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-3,1,1).
FORMULA
a(n)=a(n-1)+3*a(n-2)-a(n-3)-3*a(n-4)+a(n-5)+a(n-6).
G.f.: -x*(x-1)*(1+x)*(x^2+x-1) / ( -1+x+3*x^2-x^3-3*x^4+x^5+x^6 ). - R. J. Mathar, May 06 2014
EXAMPLE
MATHEMATICA
q = x^3; s = x^2 + 1; z = 40;
p[n_, x_] := Fibonacci[n, x];
Table[Expand[p[n, x]], {n, 1, 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, 1, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
(* A192780 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
(* A192781 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
(* A192782 *)
LinearRecurrence[{1, 3, -1, -3, 1, 1}, {1, 0, 1, 1, 2, 5}, 40] (* Harvey P. Dale, Nov 07 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 09 2011
STATUS
approved