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A154247 a(n) = ( (6 + sqrt(7))^n - (6 - sqrt(7))^n )/(2*sqrt(7)).

%I A154247 #37 Sep 08 2022 08:45:40
%S A154247 1,12,115,1032,9049,78660,681499,5896848,50998705,440975868,
%T A154247 3812747971,32964675480,285006414601,2464101386292,21304030612075,
%U A154247 184189427142432,1592456237959009,13767981468377580,119034546719719699
%N A154247 a(n) = ( (6 + sqrt(7))^n - (6 - sqrt(7))^n )/(2*sqrt(7)).
%C A154247 Lim_{n -> infinity} a(n)/a(n-1) = 6 + sqrt(7) = 8.6457513110....
%H A154247 G. C. Greubel, <a href="/A154247/b154247.txt">Table of n, a(n) for n = 1..750</a>
%H A154247 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12, -29).
%F A154247 From _Philippe Deléham_, Jan 06 2009: (Start)
%F A154247 a(n) = 12*a(n-1) - 29*a(n-2) for n > 1, with a(0)=0, a(1)=1.
%F A154247 G.f.: x/(1 - 12*x + 29*x^2). (End)
%F A154247 E.g.f.: sinh(sqrt(7)*x)*exp(6*x)/sqrt(7). - _Ilya Gutkovskiy_, Sep 08 2016
%t A154247 Join[{a=1,b=12},Table[c=12*b-29*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 31 2011 *)
%t A154247 With[{c=Sqrt[7]},Simplify/@Table[((6+c)^n-(6-c)^n)/(2c),{n,20}]] (* or *) LinearRecurrence[{12,-29},{1,12},20] (* _Harvey P. Dale_, Mar 02 2012 *)
%o A154247 (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-7); S:=[ ((6+r)^n-(6-r)^n)/(2*r): n in [1..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jan 07 2009
%o A154247 (Magma) I:=[1,12]; [n le 2 select I[n] else 12*Self(n-1)-29*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Sep 08 2016
%o A154247 (Sage) [lucas_number1(n,12,29) for n in range(1, 20)] # _Zerinvary Lajos_, Apr 27 2009
%Y A154247 Cf. A010465 (decimal expansion of square root of 7).
%K A154247 nonn
%O A154247 1,2
%A A154247 Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009
%E A154247 Extended beyond a(7) by _Klaus Brockhaus_, Jan 07 2009
%E A154247 Edited by _Klaus Brockhaus_, Oct 06 2009