A320918 - OEIS

%I #26 Dec 10 2023 17:23:51

%S 3,-9,41,-186,845,-3844,17510,-79865,364741,-1667859,7636046,

%T -35002493,160633658,-738017016,3394477491,-15629323441,72036344133,

%U -332346150886,1534759151873,-7093873005004,32817327856690,-151943731458257,704053152985509,-3264786419847751

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

%C In general, for integer h, k let

%C   X = (sin^(h+k)(2*Pi/7))/(sin^(h)(4*Pi/7)*sin^(k)(8*Pi/7)),

%C   Y = (sin^(h+k)(4*Pi/7))/(sin^(h)(8*Pi/7)*sin^(k)(2*Pi/7)),

%C   Z = (sin^(h+k)(8*Pi/7))/(sin^(h)(2*Pi/7)*sin^(k)(4*Pi/7)).

%C then X, Y, Z are the roots of a monic equation

%C     t^3 + a*t^2 + b*t + c = 0

%C where a, b, c are integers and c = 1 or -1.

%C Then X^n + Y^n + Z^n , n = 0, 1, 2, ... is an integer sequence.

%C Instances of such sequences with (h,k) values:

%C   (-3,0), (0,3), (3,-3): gives A274663;

%C   (-3,3), (0,-3): give A274664;

%C   (-2,0), (0,2), (2,-2): give A198636;

%C   (-2,-3), (-1,-2), (2,-1), (3,-1): give A274032;

%C   (-1,-1), (-1,2): give A215076;

%C   (-1,0), (0,1), (1,-1): give A094648;

%C   (-1,1), (0,-1), (1,0):  give A274975;

%C   (1,1), (-2,1), (1,-2): give A274220;

%C   (1,2), (-3,1), (2,-3: give A274075;

%C   (1,3): this sequence.

%H Colin Barker, <a href="/A320918/b320918.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-9,-20,1).

%F a(n) = ((sin^4(2*Pi/7))/(sin(4*Pi/7)*sin^3(8*Pi/7)))^n

%F      + ((sin^4(4*Pi/7))/(sin(8*Pi/7)*sin^3(2*Pi/7)))^n

%F      + ((sin^4(8*Pi/7))/(sin(2*Pi/7)*sin^3(4*Pi/7)))^n.

%F a(n) = -9*a(n-1) - 20*a(n-2) + a(n-3) for n>2.

%F G.f.: (3 + 18*x + 20*x^2) / (1 + 9*x + 20*x^2 - x^3). - _Colin Barker_, Dec 09 2018

%p a := proc(n) option remember; if n < 3 then [3, -9, 41][n+1] else

%p -9*a(n-1) - 20*a(n-2) + a(n-3) fi end: seq(a(n), n=0..32); # _Peter Luschny_, Oct 25 2018

%t 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 *)

%t LinearRecurrence[{-9,-20,1},{3,-9,41},30] (* _Harvey P. Dale_, Dec 10 2023 *)

%o (PARI) polsym(x^3 + 9*x^2 + 20*x - 1, 25) \\ _Joerg Arndt_, Oct 24 2018

%o (PARI) Vec((3 + 18*x + 20*x^2) / (1 + 9*x + 20*x^2 - x^3) + O(x^30)) \\ _Colin Barker_, Dec 09 2018

%Y Cf. A248417, A274032, A274075.

%K sign,easy

%O 0,1

%A _Kai Wang_, Oct 24 2018