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A014445
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Even Fibonacci numbers; or, Fibonacci(3n).
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23
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0, 2, 8, 34, 144, 610, 2584, 10946, 46368, 196418, 832040, 3524578, 14930352, 63245986, 267914296, 1134903170, 4807526976, 20365011074, 86267571272, 365435296162, 1548008755920, 6557470319842, 27777890035288, 117669030460994
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 232.
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.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
Tanya Khovanova, Recursive Sequences
R. Knott, Mathematics of the Fibonacci Series
Index entries for sequences related to linear recurrences with constant coefficients, signature (4,1)
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FORMULA
| a(n)=sum(k=0, n, binomial(n, k)*F(k)*2^k) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 25 2003
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
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
a(n)=sum(k=0, n, F(n+k)*binomial(n, k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 15 2005
O.g.f.: -2*x/(-1+4*x+x^2). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 06 2008
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. [From Al Hakanson (hawkuu(AT)gmail.com), May 02 2009]
a(n)= 3*F(n-1)*F(n)*F(n+1) +2*F(n)^3, F(n)=A000045(n) [From Gary Detlefs (gdetlefs(AT)aol.com) Dec 23 2010]
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MAPLE
| (Mupad) numlib::fibonacci(3*n) $ n = 0..30; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2008
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MATHEMATICA
| Table[Fibonacci[3n], {n, 0, 23}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 07 2006
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PROG
| (Other) sage: [fibonacci(3*n) for n in xrange(0, 24)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]
(MAGMA) [Fibonacci(3*n): n in [0..50]]; // Vincenzo Librandi, Apr 18 2011
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CROSSREFS
| Cf. A000045, A001076.
First differences of A099919. Third column of array A102310.
Sequence in context: A000163 A117616 A192402 * A113440 A034999 A067336
Adjacent sequences: A014442 A014443 A014444 * A014446 A014447 A014448
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Mohammad K. Azarian (ma3(AT)evansville.edu)
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EXTENSIONS
| More terms from Jud McCranie, JudMcCranie(AT)ugaalum.uga.edu.
One more term from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 07 2006
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