OFFSET
0,1
COMMENTS
In general, for integer h, k let
X = (sin^(h+k)(2*Pi/7))/(sin^(h)(4*Pi/7)*sin^(k)(8*Pi/7)),
Y = (sin^(h+k)(4*Pi/7))/(sin^(h)(8*Pi/7)*sin^(k)(2*Pi/7)),
Z = (sin^(h+k)(8*Pi/7))/(sin^(h)(2*Pi/7)*sin^(k)(4*Pi/7)).
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.
Instances of such sequences with (h,k) values:
(-3,0), (0,3), (3,-3): gives A274663;
(-3,3), (0,-3): give A274664;
(-2,0), (0,2), (2,-2): give A198636;
(-2,-3), (-1,-2), (2,-1), (3,-1): give A274032;
(-1,-1), (-1,2): give A215076;
(-1,0), (0,1), (1,-1): give A094648;
(-1,1), (0,-1), (1,0): give A274975;
(1,1), (-2,1), (1,-2): give A274220;
(1,2), (-3,1), (2,-3: give A274075;
(1,3): this sequence.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-9,-20,1).
FORMULA
a(n) = ((sin^4(2*Pi/7))/(sin(4*Pi/7)*sin^3(8*Pi/7)))^n
+ ((sin^4(4*Pi/7))/(sin(8*Pi/7)*sin^3(2*Pi/7)))^n
+ ((sin^4(8*Pi/7))/(sin(2*Pi/7)*sin^3(4*Pi/7)))^n.
a(n) = -9*a(n-1) - 20*a(n-2) + a(n-3) for n>2.
G.f.: (3 + 18*x + 20*x^2) / (1 + 9*x + 20*x^2 - x^3). - Colin Barker, Dec 09 2018
MAPLE
a := proc(n) option remember; if n < 3 then [3, -9, 41][n+1] else
-9*a(n-1) - 20*a(n-2) + a(n-3) fi end: seq(a(n), n=0..32); # Peter Luschny, Oct 25 2018
MATHEMATICA
CoefficientList[Series[(3 + 18*x + 20*x^2) / (1 + 9*x + 20*x^2 - x^3) , {x, 0, 50}], x] (* Amiram Eldar, Dec 09 2018 *)
LinearRecurrence[{-9, -20, 1}, {3, -9, 41}, 30] (* Harvey P. Dale, Dec 10 2023 *)
PROG
(PARI) polsym(x^3 + 9*x^2 + 20*x - 1, 25) \\ Joerg Arndt, Oct 24 2018
(PARI) Vec((3 + 18*x + 20*x^2) / (1 + 9*x + 20*x^2 - x^3) + O(x^30)) \\ Colin Barker, Dec 09 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Kai Wang, Oct 24 2018
STATUS
approved