Search: id:A006497
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%I A006497 M0910
%S A006497 2,3,11,36,119,393,1298,4287,14159,46764,154451,510117,1684802,5564523,
%T A006497 18378371,60699636,200477279,662131473,2186871698,7222746567,
%U A006497 23855111399,78788080764,260219353691,859446141837,2838557779202
%N A006497 a(n) = 3a(n-1) + a(n-2).
%D A006497 A. F. Horadam, Generating identities for generalized Fibonacci and Lucas 
               triples, Fib. Quart., 15 (1977), 289-292.
%D A006497 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A006497 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A006497 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A006497 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A006497 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A006497 <a href="Sindx_Rea.html#recur1">Index entries for recurrences a(n) = 
               k*a(n - 1) +/- a(n - 2)</a>
%H A006497 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A006497 a(n) = [(3 + sqrt13)/2]^n + [(3 - sqrt13)/2]^n; A006190(n-2) + A006190(n) 
               = a(n-1); [a(n)]^2 - 13[A006190(n)]^2 = 4(-1)^n. - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Jun 15 2003
%F A006497 E.g.f. : 2exp(3x/2)cosh(sqrt(13)x/2); a(n)=2^(1-n)sum{k=0..floor(n/2), 
               C(n, 2k)13^k3^(n-2k)}. a(n)=2T(n, 3i/2)(-i)^n with T(n, x) Chebyshev's 
               polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry 
               (pbarry(AT)wit.ie), Nov 15 2003
%F A006497 Comments from Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jan 
               02 2009 (Start): fract(((3+sqrt(13))/2)^n))=(1/2)*(1+(-1)^n)-(-1)^n*((3+sqrt(13))/
               2)^(-n)=(1/2)*(1+(-1)^n)-((3-sqrt(13))/2)^n.
%F A006497 See A001622 for a general formula concerning the fractional parts of 
               powers of numbers x>1, which suffice x-x^(-1)=floor(x).
%F A006497 a(n) = nint(((3+sqrt(13))/2)^n) for n>0. (End)
%p A006497 A006497:=(-2+3*z)/(-1+3*z+z**2); [S. Plouffe in his 1992 dissertation.]
%t A006497 Table[LucasL[n, 3], {n, 0, 26}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jul 09 2009]
%o A006497 (Other) sage: [lucas_number2(n,3,-1) for n in xrange(0, 25)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
%Y A006497 Cf. A006190.
%Y A006497 Cf. A100230.
%Y A006497 Cf. A001622, A014176, A080039, A098316.
%Y A006497 Sequence in context: A144056 A062630 A159458 this_sequence A038912 A019361 
               A093804
%Y A006497 Adjacent sequences: A006494 A006495 A006496 this_sequence A006498 A006499 
               A006500
%K A006497 nonn,easy
%O A006497 0,1
%A A006497 N. J. A. Sloane (njas(AT)research.att.com).
%E A006497 More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Feb 14 
               2004

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