A320918 Sum of n-th powers of the roots of x^3 + 9*x^2 + 20*x - 1.

3, -9, 41, -186, 845, -3844, 17510, -79865, 364741, -1667859, 7636046, -35002493, 160633658, -738017016, 3394477491, -15629323441, 72036344133, -332346150886, 1534759151873, -7093873005004, 32817327856690, -151943731458257, 704053152985509, -3264786419847751

Offset: 0

Author

Kai Wang, Oct 24 2018

Keywords

• sign
• easy

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).

Crossrefs

Cf. A248417, A274032, A274075.

Programs

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 *)

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

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