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 A322461 Sum of n-th powers of the roots of x^3 + 8*x^2 + 5*x - 1. 1
 3, -8, 54, -389, 2834, -20673, 150825, -1100401, 8028410, -58574450, 427353149, -3117924532, 22748056061, -165967472679, 1210881576595, -8834467193304, 64455362190778, -470259679983109, 3430966161717678, -25031975531635101, 182630713764509309 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Let A = cos(2*Pi/7), B = cos(4*Pi/7), C = cos(8*Pi/7). For integers h, k let    X = 2*sqrt(7)*A^(h+k+1)/(B^h*C^k),    Y = 2*sqrt(7)*B^(h+k+1)/(C^h*A^k),    Z = 2*sqrt(7)*C^(h+k+1)/(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) = (0,1). LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (-8,-5,1). FORMULA a(n) = (2*sqrt(7)*A^2/C)^n + (2*sqrt(7)*B^2/A)^n + (2*sqrt(7)*C^2/B)^n, where A = cos(2*Pi/7), B = cos(4*Pi/7), C = cos(8*Pi/7). a(n) = -8*a(n-2) - 5*a(n-2) + a(n-3) for n > 2. G.f.: (3 + x)*(1 + 5*x) / (1 + 8*x + 5*x^2 - x^3). - Colin Barker, Dec 09 2018 MATHEMATICA LinearRecurrence[{-8, -5, 1}, {3, -8, 54}, 50] (* Amiram Eldar, Dec 09 2018 *) PROG (PARI) Vec((3 + x)*(1 + 5*x) / (1 + 8*x + 5*x^2 - x^3) + O(x^25)) \\ Colin Barker, Dec 09 2018 (PARI) polsym(x^3 + 8*x^2 + 5*x - 1, 25) \\ Joerg Arndt, Dec 17 2018 CROSSREFS Similar sequences with (h,k) values: A094648 (0,0), A274075 (1,0). Sequence in context: A153385 A026088 A354851 * A192647 A121567 A222788 Adjacent sequences:  A322458 A322459 A322460 * A322462 A322463 A322464 KEYWORD sign,easy AUTHOR Kai Wang, Dec 09 2018 STATUS approved

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Last modified June 26 10:23 EDT 2022. Contains 354879 sequences. (Running on oeis4.)