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A014445 Even Fibonacci numbers; or, Fibonacci(3n). 36
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 the papers of Witula et al.). - 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 to Fibonacci Quart. 2012).

LINKS

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

Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876 (See Corollary 1 (v)).

I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5

Tanya Khovanova, Recursive Sequences

Ron Knott, Mathematics of the Fibonacci Series

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

R. Witula, Binomials transformations formulas of scaled Lucas numbers, Demonstratio Math. 46:1 (2013), pp. 15-27.

Index entries for 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_{k=0..floor((n-1)/2)} C(n, 2k+1)*5^k*2^(n-2k). - 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+sqrt(5))^n - (2-sqrt(5))^n)/sqrt(20). - 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). - 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

For n > 0, a(n) = 5*F(n-1)*F(n)*F(n+1) - 2*F(n)*(-1)^n. - J. M. Bergot, Dec 10 2015

a(n) = -(-1)^n * a(-n) for all n in Z. - Michael Somos, Nov 15 2018

EXAMPLE

G.f. = 2*x + 8*x^2 + 34*x^3 + 144*x^4 + 610*x^5 + 2584*x^6 + 10946*x^7 + ...

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)] # 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 A296227 A034999

Adjacent sequences:  A014442 A014443 A014444 * A014446 A014447 A014448

KEYWORD

nonn,easy,nice

AUTHOR

Mohammad K. Azarian

STATUS

approved

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Last modified December 16 20:51 EST 2018. Contains 318189 sequences. (Running on oeis4.)