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A033888 a(n) = Fibonacci(4n). 23
0, 3, 21, 144, 987, 6765, 46368, 317811, 2178309, 14930352, 102334155, 701408733, 4807526976, 32951280099, 225851433717, 1548008755920, 10610209857723, 72723460248141, 498454011879264, 3416454622906707, 23416728348467685, 160500643816367088, 1100087778366101931, 7540113804746346429 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

(x,y)=(a(n),a(n+1)) are solutions of (x+y)^2/(1+xy)=9, the other solutions are in A033890. - Floor van Lamoen, Dec 10 2001

Sequence A033888 provides half of the solutions to the equation 5*x^2 + 4 is a square. The other half are found in A033890. Lim. n-> Inf. a(n)/a(n-1) = phi^4 = (7+3*Sqrt(5))/2. - Gregory V. Richardson, Oct 13 2002

Fibonacci numbers divisible by 3. - Reinhard Zumkeller, Aug 20 2011

LINKS

Nathaniel Johnston, Table of n, a(n) for n = 0..300

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (7,-1).

FORMULA

a(n) = 7*a(n-1) - a(n-2).

a(n) = [(7+3*sqrt(5))^(n-1) - (7-3*sqrt(5))^(n-1)] / ((2^(n-1))*sqrt(5)). - Gregory V. Richardson, Oct 13 2002

a(n) = Sum_{k=0..n} F(3n-k)*binomial(n, k). - Benoit Cloitre, Jun 07 2004

Lucas(2n) * Lucas(n) * Fibonacci(n). - Ralf Stephan, Sep 25 2004

G.f.: 3*x/(1-7*x+x^2). - Philippe Deléham, Nov 17 2008

a(n) = 3*A004187(n). - R. J. Mathar, Sep 03 2010

a(n) Fibonacci[(8 n + 5)] modulo Fibonacci[(8 n + 1)]. -Artur Jasinski, Nov 15 2011

EXAMPLE

G.f. = 3*x + 21*x^2 + 144*x^3 + 987*x^4 + 6765*x^5 + 46368*x^6 + ...

MAPLE

A033888:=n->combinat[fibonacci](4*n): seq(A033888(n), n=0..30); # Wesley Ivan Hurt, Apr 26 2017

MATHEMATICA

Table[Fibonacci[4*n], {n, 0, 14}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2008 *)

Table[Mod[Fibonacci[(8 n + 5)] , Fibonacci[(8 n + 1)]], {n, 1, 10}] (* Artur Jasinski, Nov 15 2011 *)

PROG

(Mupad) numlib::fibonacci(n*4) $ n = 0..30; - Zerinvary Lajos, May 08 2008

(Sage) [lucas_number1(n, 3, 1)*lucas_number2(n, 3, 1) for n in xrange(0, 21)] # Zerinvary Lajos, Jun 28 2008

(Sage) [fibonacci(4*n) for n in xrange(0, 20)] # Zerinvary Lajos, May 15 2009

(MAGMA) [ Fibonacci(4*n): n in [0..100]]; // Vincenzo Librandi, Apr 15 2011

(PARI) a(n)=fibonacci(4*n) \\ Charles R Greathouse IV, Feb 03 2014

CROSSREFS

Cf. A000045.

Fourth column of array A102310.

Sequence in context: A137969 A054419 A228115 * A141492 A243397 A173350

Adjacent sequences:  A033885 A033886 A033887 * A033889 A033890 A033891

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 21 16:08 EST 2017. Contains 295003 sequences.