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A014445 Even Fibonacci numbers; or, Fibonacci(3n). 30
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; text; internal format)
OFFSET

0,2

COMMENTS

a(n) = 3^n*b(n;2/3) = -b(n;-2), but we have 3^n*a(n;2/3) = F(3n+1) = A033887 and a(n;-2) = F(3n-1) = A015448, where a(n;d) and b(n;d), n=0,1,..., d, denote the, so called, delta-Fibonacci numbers (the argument "d" of a(n;d) and b(n;d) is abbreviation of the symbol "delta") defined by the following equivalent relations: (1 + d*((sqrt(5) - 1)/2))^n = a(n;d) + b(n;d)*((sqrt(5) - 1)/2) equiv. a(0;d)=1, b(0;d)=0, a(n+1;d)=a(n;d)+d*b(n;d), b(n+1;d)=d*a(n;d)+(1-d)b(n;d) equiv. a(0;d)=a(1;d)=1, b(0;1)=0, b(1;d)=d, and x(n+2;d) + (d-2)*x(n+1;d) + (1-d-d^2)*x(n;d) = 0 for every n=0,1,...,d, and x=a,b equiv. a(n;d) = sum_{k=0,..,n} C(n,k)*F(k-1)*(-d)^k, and b(n;d) = sum_{k=0,..,n} C(n,k)*(-1)^(k-1)*F(k)*d^k equiv. a(n;d) = sum_{k=0,..,n} C(n,k)*F(k+1)*(1-d)^(n-k)*d^k, and b(n;d) = sum_{k=1,..,n} C(n;k)*F(k)*(1-d)^(n-k)*d^k. The sequences a(n;d) and b(n;d) for special values d are connected with many known sequences: A000045, A001519, A001906, A015448, A020699, A033887, A033889, A074872, A081567, A081568, A081569, A081574, A081575, A163073 (see also Witula's et al. papers). - Roman Witula, Jul 12 2012

For any odd k, Fibonacci(k*n)= sqrt(Fibonacci((k-1)*n)*Fibonacci((k+1)*n)+Fibonacci(n)^2). [Gary Detlefs, Dec 28 2012]

REFERENCES

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 232.

D. Chmiela, K. Kaczmarek, R. Witula, Binomials Transformation Formulae of Scaled Fibonacci Numbers,  (submitted 2012).

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.

R. Witula, Binomials Transformations Formulae of Scaled Lucas Numbers, Demonstratio Math. (in press, 2012).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

Tanya Khovanova, Recursive Sequences

R. Knott, Mathematics of the Fibonacci Series

R. Witula, Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009) 310-329, MR2555042

Index entries for sequences related to linear recurrences with constant coefficients, signature (4,1)

FORMULA

a(n)=sum(k=0, n, binomial(n, k)*F(k)*2^k) - Benoit Cloitre, 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, 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, May 15 2005

O.g.f.: -2*x/(-1+4*x+x^2). - R. J. Mathar, 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 Dec 23 2010]

a(n) = (-1)^n*3*F(n) + 5*F(n)^3, n>=0. See the D. Jennings formula given in a comment on A111125, where also the reference is given. - Wolfdieter Lang, Aug 31 2012

With L(n) a Lucas number, F(3n)=F(n)*(L(2n)+(-1)^n)=(L(3n+1)+L(3n-1))/5 starting at n=1. - J. M. Bergot, Oct 25 2012

a(n) = sqrt(Fibonacci(2*n)*Fibonacci(4*n)+Fibonacci(n)^2). [Gary Detlefs, Dec 28 2012]

MAPLE

(Mupad) numlib::fibonacci(3*n) $ n = 0..30; - Zerinvary Lajos, May 09 2008

MATHEMATICA

Table[Fibonacci[3n], {n, 0, 23}] - Stefan Steinerberger, Apr 07 2006

PROG

(Sage) [fibonacci(3*n) for n in xrange(0, 24)]# [From Zerinvary Lajos, May 15 2009]

(MAGMA) [Fibonacci(3*n): n in [0..50]]; // Vincenzo Librandi, Apr 18 2011

(PARI) a(n)=fibonacci(3*n) \\ Charles R Greathouse IV, Oct 25 2012

CROSSREFS

Cf. A000045, A001076.

First differences of A099919. Third column of array A102310.

Sequence in context: A117616 A228655 A192402 * A113440 A034999 A067336

Adjacent sequences:  A014442 A014443 A014444 * A014446 A014447 A014448

KEYWORD

nonn,easy,nice

AUTHOR

Mohammad K. Azarian

EXTENSIONS

More terms from Jud McCranie.

One more term from Stefan Steinerberger, Apr 07 2006

STATUS

approved

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Last modified July 31 00:52 EDT 2014. Contains 245078 sequences.