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 A188599 Expansion of x/(1-6*x+25*x^2). 3
 0, 1, 6, 11, -84, -779, -2574, 4031, 88536, 430441, 369246, -8545549, -60504444, -149387939, 616283466, 7432399271, 29187308976, -10686127919, -793799491914, -4495643753509, -7128875223204, 69617842498501, 595928935571106, 1835127550964111, -3887458083492984 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Original name: G.f.: 1/(1-6*x+25*x^2). Suggested by Philippe Flajolet as an example of a simple formula for which the general term is hard to guess because 1-6*x+25*x^2 has 2 complex roots of equal size and modulus 1. The Lucas sequence U_n(6,25). - Peter Bala, Feb 02 2017 REFERENCES Discussion in 1993 at the FPSAC 1993 in Florence. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 110 terms from Vincenzo Librandi) Wikipedia, Lucas sequence Index entries for linear recurrences with constant coefficients, signature (6,-25). FORMULA a(n) = ((3+4*i)^n-(3-4*i)^n)/8/i, where i=sqrt(-1). - Denis Excoffier, Jan 19 2013 From Peter Bala, Feb 02 2017: (Start) a(n) = (1/4)*( Re((2 - i)^n)*Im((2 + i)^n) - Re((2 + i)^n)*Im((2 - i)^n) ). a(n) = (1/2) * the directed or signed area of the triangle in the complex plane with vertices at the points 0, (2 - i)^n and (2 + i)^n. (End) a(n) = 5^n*sin(n*arctan(1/2))*cos(n*arctan(1/2))/2. - Peter Luschny, Feb 02 2017 E.g.f.: (1/4)*exp(3*x)*sin(4*x). - Stefano Spezia, Feb 01 2020 MAPLE x/(1-6*x+25*x^2):series(%, x, 44):seriestolist(%); MATHEMATICA Table[Im[(3 + 4*I)^n]/4, {n, 0, 22}] (* Jean-François Alcover, Jun 14 2011 *) CoefficientList[Series[x/(1-6*x+25*x^2), {x, 0, 30}], x] (* Harvey P. Dale, Dec 01 2018 *) LinearRecurrence[{6, -25}, {0, 1}, 30] (* Harvey P. Dale, Jul 03 2021 *) PROG (PARI) Vec(x/(1-6*x+25*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 14 2011 CROSSREFS Sequence in context: A219702 A013321 A192893 * A217934 A012419 A012663 Adjacent sequences:  A188596 A188597 A188598 * A188600 A188601 A188602 KEYWORD sign,easy AUTHOR Simon Plouffe, Apr 06 2011 EXTENSIONS Minor edits by N. J. A. Sloane, Apr 06 2011 Minor modification to Name by Peter Bala, Feb 02 2017 STATUS approved

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Last modified May 28 00:30 EDT 2022. Contains 354110 sequences. (Running on oeis4.)