Search: id:A014445
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%I A014445
%S A014445 0,2,8,34,144,610,2584,10946,46368,196418,832040,3524578,14930352,
%T A014445 63245986,267914296,1134903170,4807526976,20365011074,86267571272,
%U A014445 365435296162,1548008755920,6557470319842,27777890035288,117669030460994
%N A014445 Even Fibonacci numbers; or, Fibonacci_{3k}.
%D A014445 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, id. 232.
%D A014445 Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the 
               Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 
               06.1.1.
%H A014445 T. D. Noe, <a href="b014445.txt">Table of n, a(n) for n=0..200</a>
%H A014445 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A014445 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A014445 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
               fibmaths.html">Mathematics of the Fibonacci Series</a>
%F A014445 a(n)=sum(k=0, n, binomial(n, k)*F(k)*2^k) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Oct 25 2003
%F A014445 a(n) = 4*a(n-1) + a(n-2); a(-1) = 2, a(0) = 0. a(n) = 2*A001076(n). a(n) 
               = (F(n+1))^3 + (F(n))^3 - (F(n-1))^3. - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Jun 11 2004
%F A014445 a(n)=Sum(C(n, 2k+1)5^k 2^(n-2k), k=0, .., Floor[(n-1)/2]) - Mario Catalani 
               (mario.catalani(AT)unito.it), Jul 22 2004
%F A014445 a(n)=sum(k=0, n, F(n+k)*binomial(n, k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               May 15 2005
%F A014445 O.g.f.: -2*x/(-1+4*x+x^2). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 06 2008
%F A014445 a(n)=second binomial transform of (2,4,10,20,50,100,250). This is 2* 
               (1,2,5,10,25,50,125) or 5^n (offset 0) *2 for the odd numbers or 
               *4 for the even. The sequences are interpolated. Also a(n)=2*((2+sqrt5)^n-(2-sqrt5)^n)/
               sqrt20 offset 1. [From Al Hakanson (hawkuu(AT)gmail.com), May 02 
               2009]
%p A014445 (Mupad) numlib::fibonacci(3*n) $ n = 0..30; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               May 09 2008
%t A014445 Table[Fibonacci[3n], {n, 0, 23}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 07 2006
%o A014445 (Other) sage: [fibonacci(3*n) for n in xrange(0, 24)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]
%Y A014445 Cf. A000045, A001076.
%Y A014445 Equals 2*A001076. First differences of A099919. Third column of array 
               A102310.
%Y A014445 Sequence in context: A111643 A000163 A117616 this_sequence A113440 A034999 
               A067336
%Y A014445 Adjacent sequences: A014442 A014443 A014444 this_sequence A014446 A014447 
               A014448
%K A014445 nonn,easy,nice
%O A014445 0,2
%A A014445 Mohammad K. Azarian (ma3(AT)evansville.edu)
%E A014445 More terms from Jud McCranie, j.mccranie(AT)comcast.net.
%E A014445 One more term from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 07 2006

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