OFFSET
0,1
COMMENTS
Let A = cos(2*Pi/7), B = cos(4*Pi/7), C = cos(8*Pi/7).
In general, for integer h, k let
X = A^(h+k)/(B^h*C^k),
Y = B^(h+k)/(C^h*A^k),
Z = C^(h+k)/(A^h*B^k).
then X, Y, Z are the roots of a monic equation
t^3 + a*t^2 + b*t + c = 0
where a, b, c are integers and c = 1 or -1.
Then X^n + Y^n + Z^n , n = 0, 1, 2, ... is an integer sequence.
This sequence has (h,k) = (1,3).
LINKS
Colin Barker, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (-95,88,1).
FORMULA
a(n) = (A^4/(B*C^3))^n + (B^4/(C*A^3))^n + (C^4/(A*B^3))^n.
a(n) = -95*a(n-1) + 88*a(n-2) + a(n-3) for n>2.
G.f.: (3 + 190*x - 88*x^2) / (1 + 95*x - 88*x^2 - x^3). - Colin Barker, Dec 09 2018
MAPLE
seq(coeff(series((3+190*x-88*x^2)/(1+95*x-88*x^2-x^3), x, n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Dec 11 2018
MATHEMATICA
LinearRecurrence[{-95, 88, 1}, {3, -95, 9201}, 50] (* Amiram Eldar, Dec 09 2018 *)
PROG
(PARI) Vec((3 + 190*x - 88*x^2) / (1 + 95*x - 88*x^2 - x^3) + O(x^15)) \\ Colin Barker, Dec 09 2018
(PARI) polsym(x^3 + 95*x^2 - 88*x - 1, 25) \\ Joerg Arndt, Dec 17 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Kai Wang, Dec 09 2018
STATUS
approved