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 A153596 a(n) = ((5 + sqrt(3))^n - (5 - sqrt(3))^n)/(2*sqrt(3)). 3
 1, 10, 78, 560, 3884, 26520, 179752, 1214080, 8186256, 55152800, 371430368, 2500942080, 16837952704, 113358801280, 763153053312, 5137636904960, 34587001876736, 232842006858240, 1567506027294208, 10552536122060800, 71040228620135424 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Third binomial transform of A054485. Fifth binomial transform of A162813 preceded by 1. Lim_{n -> infinity} a(n)/a(n-1) = 5 + sqrt(3) = 6.73205080756887729.... LINKS G. C. Greubel, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (10,-22). FORMULA G.f.: x/(1 - 10*x + 22*x^2). - Klaus Brockhaus, Dec 31 2008 [corrected Oct 11 2009] a(n) = 10*a(n-1) - 22*a(n-2) for n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009 E.g.f.: sinh(sqrt(3)*x)*exp(5*x)/sqrt(3). - Ilya Gutkovskiy, Aug 23 2016 MATHEMATICA Table[Simplify[((5+Sqrt[3])^n -(5-Sqrt[3])^n)/(2*Sqrt[3])], {n, 1, 25}] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011, modified by G. C. Greubel, Jun 01 2019 *) LinearRecurrence[{10, -22}, {1, 10}, 25] (* G. C. Greubel, Aug 22 2016 *) PROG (Magma) Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((5+r)^n-(5-r)^n)/(2*r): n in [1..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008 (Sage) [lucas_number1(n, 10, 22) for n in range(1, 25)] # Zerinvary Lajos, Apr 26 2009 (Magma) I:=[1, 10]; [n le 2 select I[n] else 10*Self(n-1)-22*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016 (PARI) my(x='x+O('x^25)); Vec(x/(1-10*x+22*x^2)) \\ G. C. Greubel, Jun 01 2019 CROSSREFS Cf. A002194 (decimal expansion of sqrt(3)), A054485, A162813. Sequence in context: A080618 A298270 A082136 * A316595 A348314 A056986 Adjacent sequences: A153593 A153594 A153595 * A153597 A153598 A153599 KEYWORD nonn AUTHOR Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008 EXTENSIONS Extended beyond a(7) by Klaus Brockhaus, Dec 31 2008 Edited by Klaus Brockhaus, Oct 11 2009 STATUS approved

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Last modified December 8 06:47 EST 2022. Contains 358673 sequences. (Running on oeis4.)