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A033888
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Fibonacci(4n).
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18
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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; internal format)
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OFFSET
| 0,2
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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 (fvlamoen(AT)hotmail.com), 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 (omomom(AT)hotmail.com), Oct 13 2002
Fibonacci numbers divisible by 3. [Reinhard Zumkeller, Aug 20 2011]
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
Nathaniel Johnston, Table of n, a(n) for n = 0..300
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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 (omomom(AT)hotmail.com), Oct 13 2002
a(n) = sum(k=0, n, F(3n-k)*binomial(n, k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 07 2004
Lucas(2n) * Lucas(n) * Fibonacci(n). - R. Stephan, Sep 25 2004
G.f.: 3x/(1-7x+x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 17 2008]
a(n) = 3*A004187(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 03 2010]
a(n) Fibonacci[(8 n + 5)] modulo Fibonacci[(8 n + 1)] [From Artur Jasinski grafix(AT)csl.pl), Nov 15 2011]
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MAPLE
| (Mupad) numlib::fibonacci(n*4) $ n = 0..30; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 08 2008
sage: [lucas_number1(n, 3, 1)*lucas_number2(n, 3, 1) for n in xrange(0, 21)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 29 2008
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MATHEMATICA
| Table[Fibonacci[4*n], {n, 0, 14}] (* Vladimir Orlovsky, Jul 21 2008 *)
Table[Mod[Fibonacci[(8 n + 5)] , Fibonacci[(8 n + 1)]], {n, 1, 10}] (* Artur Jasinski , Nov 15 2011 *)
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PROG
| sage: [lucas_number1(n, 3, 1)*lucas_number2(n, 3, 1) for n in xrange(0, 21)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 28 2008
(Other) sage: [fibonacci(4*n) for n in xrange(0, 20)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]
(MAGMA) [ Fibonacci(4*n): n in [0..100]]; // Vincenzo Librandi, Apr 15 2011
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CROSSREFS
| Cf. A000045.
Fourth column of array A102310.
Sequence in context: A079753 A137969 A054419 * A141492 A173350 A169634
Adjacent sequences: A033885 A033886 A033887 * A033889 A033890 A033891
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| a(20) - a(23) from Vincenzo Librandi, Apr 15 2011
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