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A058635
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(2^n)-th Fibonacci number.
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9
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1, 1, 3, 21, 987, 2178309, 10610209857723, 251728825683549488150424261, 141693817714056513234709965875411919657707794958199867, 44893845313309942978077298160660626646181883623886239791269694466661322268805744081870933775586567858979269
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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REFERENCES
| Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002, p. 446.
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LINKS
| H. Hu, Z.-W. Sun and J.-X. Liu, Reciprocal sums of second order recurrent sequences, Fib. Quart. 39(2001), no. 3, 214-220.
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FORMULA
| a(n) = a(n-1)*A001566(n-2) - Joe Keane (jgk(AT)jgk.org), May 31 2002
Sum(n>=0, 1/a(n)) = (1/2)*(7-sqrt(5)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 26 2003
1/phi^2 = (.6180339...)^2 = 2/(3+sqrt5) = Sum (2 through infinity) 1/a(n) = 1/3 + 1/21 + 1/987 + 1/2178309... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 12 2003
a(n) = (G^(2^n) - (1 - G)^(2^n))/Sqrt[5] where G = GoldenRatio = (1 + Sqrt[5])/2 [From Artur Jasinski (grafix(AT)csl.pl), Oct 05 2008]
a(n)=(4/5)^(1/2)*Cosh[(2^n)*ArcCosh[((5/4)^(1/2))]] [From Artur Jasinski (grafix(AT)csl.pl), Oct 05 2008]
a(n) = (a(n-1)^3 / a(n-2)^2 + 5 * a(n-1) * a(n-2)^2) / 2, for n > 1. [From Lee A. Newberg (integer(AT)quantconsulting.com), Jul 20 2010]
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MATHEMATICA
| Table[ Fibonacci[ 2^n ], {n, 0, 9} ]
G = (1 + Sqrt[5])/2; Table[Expand[(G^(2^n) - (1 - G)^(2^n))/Sqrt[5]], {n, 1, 7}] [From Artur Jasinski (grafix(AT)csl.pl), Oct 05 2008]
Table[Round[(4/5)^(1/2)*Cosh[2^n*ArcCosh[((5/4)^(1/2))]]], {n, 1, 10}] [From Artur Jasinski (grafix(AT)csl.pl), Oct 05 2008]
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CROSSREFS
| Cf. A000045, A054783, A001566.
Sequence in context: A111433 A111435 A111438 * A077260 A012110 A054739
Adjacent sequences: A058632 A058633 A058634 * A058636 A058637 A058638
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KEYWORD
| nonn
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AUTHOR
| Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 16 2001
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