OFFSET
0,1
COMMENTS
Let A = sin(2*Pi/7), B = sin(4*Pi/7), C = sin(8*Pi/7).
In general, for integer h, k let
X = sqrt(7)*A^(h+k-1)/(2*B^h*C^k),
Y = sqrt(7)*B^(h+k-1)/(2*C^h*A^k),
Z = sqrt(7)*C^(h+k-1)/(2*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) = (1,1).
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-7,-14,-7).
FORMULA
a(n) = (sqrt(7))^n*( (A/(2*B*C))^n + (B/(2*C*A))^n + (C/(2*A*B))^n ).
a(n) = -7*a(n-1) - 14*a(n-2) - 7*a(n-3) for n>2.
G.f.: (3 + 14*x + 14*x^2) / (1 + 7*x + 14*x^2 + 7*x^3). - Colin Barker, Dec 09 2018
MATHEMATICA
LinearRecurrence[{-7, -14, -7}, {3, -7, 21}, 50] (* Amiram Eldar, Dec 09 2018 *)
CoefficientList[Series[(3+14*x+14*x^2)/(1+7*x+14*x^2+7*x^3), {x, 0, 25}], x] (* G. C. Greubel, Dec 16 2018 *)
PROG
(PARI) Vec((3 + 14*x + 14*x^2) / (1 + 7*x + 14*x^2 + 7*x^3) + O(x^40)) \\ Colin Barker, Dec 09 2018
(PARI) polsym(x^3 + 7*x^2 + 14*x + 7, 25) \\ Joerg Arndt, Dec 17 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Kai Wang, Dec 09 2018
STATUS
approved