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 A153597 a(n) = ((6 + sqrt(3))^n - (6 - sqrt(3))^n)/(2*sqrt(3)). 2
 1, 12, 111, 936, 7569, 59940, 469503, 3656016, 28378593, 219894588, 1702241487, 13170376440, 101870548209, 787824155988, 6092161780959, 47107744223904, 364251591915201, 2816463543593580, 21777259989921327, 168383822940467784 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Fourth binomial transform of A055845. lim_{n -> infinity} a(n)/a(n-1) = 6 + sqrt(3) = 7.73205080756887729.... LINKS G. C. Greubel, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (12,-33). FORMULA G.f.: x/(1 - 12*x + 33*x^2). - Klaus Brockhaus, Dec 31 2008, (corrected Oct 11 2009) a(n) = 12*a(n-1) - 33*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(6*x)/sqrt(3). - Ilya Gutkovskiy, Aug 23 2016 MATHEMATICA LinearRecurrence[{12, -33}, {1, 12}, 25] (* G. C. Greubel, Aug 22 2016 *) PROG (MAGMA) Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((6+r)^n-(6-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008 (Sage) [lucas_number1(n, 12, 33) for n in range(1, 21)] # Zerinvary Lajos, Apr 27 2009 (MAGMA) I:=[1, 12]; [n le 2 select I[n] else 12*Self(n-1)-33*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016 CROSSREFS Cf. A002194 (decimal expansion of sqrt(3)), A055845. Sequence in context: A075231 A085773 A066042 * A036733 A253091 A123933 Adjacent sequences:  A153594 A153595 A153596 * A153598 A153599 A153600 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 May 15 02:48 EDT 2021. Contains 343909 sequences. (Running on oeis4.)