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 A322459 Sum of n-th powers of the roots of x^3 + 7*x^2 + 14*x + 7. 1
 3, -7, 21, -70, 245, -882, 3234, -12005, 44933, -169099, 638666, -2417807, 9167018, -34790490, 132119827, -501941055, 1907443237, -7249766678, 27557748813, -104759610858, 398257159370, -1514069805269, 5756205681709, -21884262613787, 83201447389466, -316323894905207 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Let A = sin(2*Pi/7), B = sin(4*Pi/7), C = sin(8*Pi/7). In general, for integer h, k let X = sqrt(7)*A^(h+k-1)/(2*B^h*C^k), Y = sqrt(7)*B^(h+k-1)/(2*C^h*A^k), Z = sqrt(7)*C^(h+k-1)/(2*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) = (1,1). LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (-7,-14,-7). FORMULA a(n) = (sqrt(7))^n*( (A/(2*B*C))^n + (B/(2*C*A))^n + (C/(2*A*B))^n ). a(n) = -7*a(n-1) - 14*a(n-2) - 7*a(n-3) for n>2. G.f.: (3 + 14*x + 14*x^2) / (1 + 7*x + 14*x^2 + 7*x^3). - Colin Barker, Dec 09 2018 MATHEMATICA LinearRecurrence[{-7, -14, -7}, {3, -7, 21}, 50] (* Amiram Eldar, Dec 09 2018 *) CoefficientList[Series[(3+14*x+14*x^2)/(1+7*x+14*x^2+7*x^3), {x, 0, 25}], x] (* G. C. Greubel, Dec 16 2018 *) PROG (PARI) Vec((3 + 14*x + 14*x^2) / (1 + 7*x + 14*x^2 + 7*x^3) + O(x^40)) \\ Colin Barker, Dec 09 2018 (PARI) polsym(x^3 + 7*x^2 + 14*x + 7, 25) \\ Joerg Arndt, Dec 17 2018 CROSSREFS Similar sequences with (h,k) values: A275831 (0,0), A215575 (0,2). Sequence in context: A240506 A037127 A105795 * A244174 A148678 A148679 Adjacent sequences: A322456 A322457 A322458 * A322460 A322461 A322462 KEYWORD sign,easy AUTHOR Kai Wang, Dec 09 2018 STATUS approved

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Last modified June 7 10:34 EDT 2023. Contains 363157 sequences. (Running on oeis4.)