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A014448 Even Lucas numbers: L(3n).

%I A014448 #87 Mar 19 2025 12:55:42
%S A014448 2,4,18,76,322,1364,5778,24476,103682,439204,1860498,7881196,33385282,
%T A014448 141422324,599074578,2537720636,10749957122,45537549124,192900153618,
%U A014448 817138163596,3461452808002,14662949395604,62113250390418
%N A014448 Even Lucas numbers: L(3n).
%C A014448 This is the Lucas sequence V(4,-1). - _Bruno Berselli_, Jan 08 2013
%H A014448 Indranil Ghosh, <a href="/A014448/b014448.txt">Table of n, a(n) for n = 0..1591</a>
%H A014448 Pooja Bhadouria, Deepika Jhala and Bijendra Singh, <a href="http://dx.doi.org/10.22436/jmcs.08.01.07">Binomial Transforms of the k-Lucas Sequences and its Properties</a>, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1 (2014), pp. 81-92; sequence L_{4,n}.
%H A014448 H. H. Ferns, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/5-2/elementary5-2.pdf">Problem B-115</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 5, No. 2 (1967), p. 202; <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/6-1/elementary6-1.pdf">Identities for F_{kn} and L{kn}</a>, Solution to Problem B-115 by Stanley Rabinowitz, ibid., Vol. 6, No. 1 (1968), pp. 92-93.
%H A014448 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A014448 Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%H A014448 Roberto Tauraso, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Tauraso/taur31.html">Some Congruences for Central Binomial Sums Involving Fibonacci and Lucas Numbers</a>, JIS 19 (2016) # 16.5.4
%H A014448 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence: Specific names</a>.
%H A014448 <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>.
%H A014448 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,1).
%F A014448 G.f.: (2-4*x)/(1-4*x-x^2).
%F A014448 a(n) = 4*a(n-1) +a(n-2) with n>1, a(0)=2, a(1)=4.
%F A014448 a(n) = (2+sqrt(5))^n + (2-sqrt(5))^n.
%F A014448 a(n) = 2*A001077(n).
%F A014448 a(n) = A000032(3*n).
%F A014448 a(n) = Sum_{k=0..n} C(n,k)*Lucas(n+k). - _Paul D. Hanna_, Oct 19 2010
%F A014448 a(n) = Fibonacci(6*n)/Fibonacci(3*n), n>0. - _Gary Detlefs_, Dec 26 2010
%F A014448 From _Peter Bala_, Mar 22 2015: (Start)
%F A014448 a(n) = ( Fibonacci(3*n + 2*k) - F(3*n - 2*k) )/Fibonacci(2*k) for nonzero integer k.
%F A014448 a(n) = ( Fibonacci(3*n + 2*k + 1) + F(3*n - 2*k - 1) )/Fibonacci(2*k + 1) for arbitrary integer k. (End)
%F A014448 a(n) = [x^n] ( (1 + 4*x + sqrt(1 + 8*x + 20*x^2))/2 )^n for n >= 1. - _Peter Bala_, Jun 23 2015
%F A014448 a(n) = L(n)*(L(n-1)*L(n+1) + 2*(-1)^n). - _J. M. Bergot_, Feb 05 2016
%F A014448 From _Peter Bala_ Oct 14 2019: (Start)
%F A014448 Sum_{n >= 1} 1/( a(n) + (-1)^(n+1)*20/a(n) ) = 3/16.
%F A014448 Sum_{n >= 1} (-1)^(n+1)/( a(n) + (-1)^(n+1)*20/a(n) ) = 1/16. (End)
%F A014448 a(n) = (15*Fibonacci(n)^2*Lucas(n) + Lucas(n)^3)/4 (Ferns, 1967). - _Amiram Eldar_, Feb 06 2022
%F A014448 E.g.f.: 2*exp(2*x)*cosh(sqrt(5)*x). - _Stefano Spezia_, Jan 18 2025
%e A014448 a(4) = L(3 * 4) = L(12) = 322. - _Indranil Ghosh_, Feb 05 2017
%t A014448 Table[LucasL[3*n], {n,0,100}] (* _G. C. Greubel_, Nov 07 2018 *)
%o A014448 (PARI) polsym(x^2-4*x-1,100)
%o A014448 (PARI) a(n)=sum(k=0,n,binomial(n,k)*(fibonacci(n+k-1)+fibonacci(n+k+1))) \\ _Paul D. Hanna_, Oct 19 2010
%o A014448 (Sage) [lucas_number2(n,4,-1) for n in range(0, 23)] # _Zerinvary Lajos_, May 14 2009
%o A014448 (Magma) [Lucas(3*n) : n in [0..100]]; // _Vincenzo Librandi_, Apr 14 2011
%Y A014448 Cf. A000032, A000045, A001077.
%Y A014448 Cf. Lucas(k*n):  A005248 (k = 2), A056854 (k = 4), A001946 (k = 5), A087215 (k = 6), A087281 (k = 7), A087265 (k = 8), A087287 (k = 9), A089772 (k = 11), A089775 (k = 12).
%K A014448 nonn,easy
%O A014448 0,1
%A A014448 _Mohammad K. Azarian_
%E A014448 More terms from _Erich Friedman_