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%I M0910 #135 Feb 14 2024 10:13:11
%S 2,3,11,36,119,393,1298,4287,14159,46764,154451,510117,1684802,
%T 5564523,18378371,60699636,200477279,662131473,2186871698,7222746567,
%U 23855111399,78788080764,260219353691,859446141837,2838557779202
%N a(n) = 3*a(n-1) + a(n-2) with a(0) = 2, a(1) = 3.
%C For more information about this type of recurrence follow the Khovanova link and see A086902 and A054413. - _Johannes W. Meijer_, Jun 12 2010
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A006497/b006497.txt">Table of n, a(n) for n = 0..1000</a>
%H P. Bhadouria, D. Jhala, and B. Singh, <a href="http://dx.doi.org/10.22436/jmcs.08.01.07">Binomial Transforms of the k-Lucas Sequences and its [sic] Properties</a>, Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92, Sequence L_{3,n}.
%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/15-4/horadam.pdf">Generating identities for generalized Fibonacci and Lucas triples</a>, Fib. Quart., 15 (1977), 289-292.
%H Haruo Hosoya, <a href="http://www.hyle.org/journal/issues/19-1/hosoya.htm">What Can Mathematical Chemistry Contribute to the Development of Mathematics?</a>, HYLE--International Journal for Philosophy of Chemistry, Vol. 19, No.1 (2013), pp. 87-105.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, and Fausto Jarquín-Zárate, <a href="https://arxiv.org/abs/1904.13002">The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d))</a>, arXiv:1904.13002 [math.NT], 2019.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,1).
%F G.f.: (2-3*x)/(1-3*x-x^2). - _Simon Plouffe_ in his 1992 dissertation
%F From _Gary W. Adamson_, Jun 15 2003: (Start)
%F a(n) = ((3 + sqrt(13))/2)^n + ((3 - sqrt(13))/2)^n. See bronze mean (A098316).
%F A006190(n-2) + A006190(n) = a(n-1).
%F a(n)^2 - 13*A006190(n)^2 = 4(-1)^n. (End)
%F From _Paul Barry_, Nov 15 2003: (Start)
%F E.g.f.: 2*exp(3*x/2)*cosh(sqrt(13)*x/2).
%F a(n) = 2^(1-n)*Sum_{k=0..floor(n/2)} C(n, 2*k)* (13)^k * 3^(n-2*k).
%F a(n) = 2*T(n, 3i/2)*(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. (End)
%F From _Hieronymus Fischer_, Jan 02 2009: (Start)
%F 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 See A001622 for a general formula concerning the fractional parts of powers of numbers x>1, which satisfy x-x^(-1)=floor(x).
%F a(n) = round(((3+sqrt(13))/2)^n) for n > 0. (End)
%F From _Johannes W. Meijer_, Jun 12 2010: (Start)
%F a(2n+1) = 3*A097783(n), a(2n) = A057076(n).
%F a(3n+1) = A041018(5n), a(3n+2) = A041018(5n+3) and a(3n+3) = 2*A041018(5n+4).
%F Limit_{k -> infinity} a(n+k)/a(k) = (a(n) + A006190(n)*sqrt(13))/2.
%F Limit_{n -> infinity} a(n)/A006190(n) = sqrt(13).
%F (End)
%F a(n) = sqrt(13*(A006190(n))^2 + 4*(-1)^n). - _Vladimir Shevelev_, Mar 13 2013
%F G.f.: G(0), where G(k) = 1 + 1/(1 - (x*(13*k-9))/((x*(13*k+4)) - 6/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 15 2013
%F a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 13*x^2))/2 )^n for n >= 1. - _Peter Bala_, Jun 23 2015
%F a(n) = Lucas(n,3), Lucas polynomials, L(n,x), evaluated at x=3. - _G. C. Greubel_, Jun 06 2019
%F a(n) = 2 * Sum_{k=0..n-2} A168561(n-2,k)*3^k + 3 * Sum_{k=0..n-1} A168561(n-1,k)*3^k, n>0. - _R. J. Mathar_, Feb 14 2024
%F a(n) = 2*A006190(n+1) - 3*A006190(n). - _R. J. Mathar_, Feb 14 2024
%p a:= n-> (<<0|1>, <1|3>>^n. <<2, 3>>)[1, 1]:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jan 26 2018
%t Table[LucasL[n, 3], {n, 0, 30}] (* _Zerinvary Lajos_, Jul 09 2009 *)
%t LucasL[Range[0, 30], 3] (* _Eric W. Weisstein_, Apr 17 2018 *)
%t LinearRecurrence[{3,1},{2,3},30] (* _Harvey P. Dale_, Feb 17 2020 *)
%o (Sage) [lucas_number2(n,3,-1) for n in range(0, 30)] # _Zerinvary Lajos_, Apr 30 2009
%o (Magma) [ n eq 1 select 2 else n eq 2 select 3 else 3*Self(n-1)+Self(n-2): n in [1..30] ]; // _Vincenzo Librandi_, Aug 20 2011
%o (Haskell)
%o a006497 n = a006497_list !! n
%o a006497_list = 2 : 3 : zipWith (+) (map (* 3) $ tail a006497_list) a006497_list
%o -- _Reinhard Zumkeller_, Feb 19 2011
%o (PARI) my(x='x+O('x^30)); Vec((2-3*x)/(1-3*x-x^2)) \\ _G. C. Greubel_, Jul 05 2017
%o (PARI) apply( {A006497(n)=[2,3]*([0,1;1,3]^n)[,1]}, [0..30]) \\ _M. F. Hasler_, Mar 06 2020
%Y Cf. A006190, A100230, A001622, A014176, A080039, A098316.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
%E Definition completed by _M. F. Hasler_, Mar 06 2020
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