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 A153594 a(n) = ((4 + sqrt(3))^n - (4 - sqrt(3))^n)/(2*sqrt(3)). 5
 1, 8, 51, 304, 1769, 10200, 58603, 336224, 1927953, 11052712, 63358307, 363181200, 2081791609, 11932977272, 68400527259, 392075513536, 2247397253921, 12882196355400, 73841406542227, 423262699717616, 2426163312691977, 13906891405206808 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Second binomial transform of A054491. Fourth binomial transform of 1 followed by A162766 and of A074324 without initial term 1. First differences are in A161728. Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(3) = 5.73205080756887729.... LINKS G. C. Greubel, Table of n, a(n) for n = 1..500 Index entries for linear recurrences with constant coefficients, signature (8,-13). FORMULA G.f.: x/(1 - 8*x + 13*x^2). - Klaus Brockhaus, Dec 31 2008, corrected Oct 11 2009 a(n) = 8*a(n-1) - 13*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(4*x)/sqrt(3). - Ilya Gutkovskiy, Aug 23 2016 a(n) = Sum_{k=0..n-1} A027907(n,2k+1)*3^k. - J. Conrad, Aug 30 2016 a(n) = Sum_{k=0..n-1} A083882(n-1-k)*4^k. - J. Conrad, Sep 03 2016 MATHEMATICA Join[{a=1, b=8}, Table[c=8*b-13*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *) LinearRecurrence[{8, -13}, {1, 8}, 40] (* Harvey P. Dale, Aug 16 2012 *) PROG (MAGMA) Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..21] ]; [ Integers()!S[j]: j in [1..#S] ];  // Klaus Brockhaus, Dec 31 2008 (Sage) [lucas_number1(n, 8, 13) for n in xrange(1, 22)] # Zerinvary Lajos, Apr 23 2009 (MAGMA) I:=[1, 8]; [n le 2 select I[n] else 8*Self(n-1)-13*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016 (PARI) a(n)=([0, 1; -13, 8]^(n-1)*[1; 8])[1, 1] \\ Charles R Greathouse IV, Sep 04 2016 CROSSREFS Cf. A002194 (decimal expansion of sqrt(3)), A054491, A074324, A161728, A162766. Sequence in context: A069325 A295348 A082135 * A316594 A037697 A037606 Adjacent sequences:  A153591 A153592 A153593 * A153595 A153596 A153597 KEYWORD nonn,easy 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 July 17 14:44 EDT 2019. Contains 325106 sequences. (Running on oeis4.)