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 A320918 Sum of n-th powers of the roots of x^3 + 9*x^2 + 20*x - 1. 2
 3, -9, 41, -186, 845, -3844, 17510, -79865, 364741, -1667859, 7636046, -35002493, 160633658, -738017016, 3394477491, -15629323441, 72036344133, -332346150886, 1534759151873, -7093873005004, 32817327856690, -151943731458257, 704053152985509, -3264786419847751 (list; graph; refs; listen; history; text; internal format)
 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 *) 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 Cf. A248417, A274032, A274075. Sequence in context: A018417 A228478 A346037 * A325289 A274739 A012246 Adjacent sequences: A320915 A320916 A320917 * A320919 A320920 A320921 KEYWORD sign,easy AUTHOR Kai Wang, Oct 24 2018 STATUS approved

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Last modified March 22 22:02 EDT 2023. Contains 361434 sequences. (Running on oeis4.)