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A194159
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Constant associated with the product of the first n nonzero even Fibonacci numbers.
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6
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8, 3, 2, 8, 8, 3, 2, 4, 4, 0, 3, 3, 9, 1, 2, 9, 8, 2, 4, 5, 0, 2, 5, 6, 6, 4, 3, 1, 3, 6, 1, 4, 2, 2, 9, 4, 2, 2, 7, 3, 2, 1, 5, 1, 9, 9, 4, 0, 9, 0, 5, 0, 3, 2, 4, 5, 1, 5, 4, 2, 2, 4, 0, 8, 9, 2, 5, 7, 6, 0, 6, 4, 8, 3, 9, 8, 5, 4, 5, 9, 9, 3, 4, 0, 8, 9, 1, 1, 6, 9, 2, 5, 6, 6, 8, 0, 5, 5, 8, 1, 8, 2, 1, 4, 9, 5, 1, 3
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OFFSET
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0,1
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COMMENTS
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The a(n) = product(F(2*i), i=1..n) is asymptotic to C2*phi^(n*(n+1))/sqrt(5)^n where phi = (1+sqrt(5))/2 and F(n) = A000045(n), see A194157. The decimal expansion of the constant C2 is given above.
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REFERENCES
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Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 6th printing with corrections. Addison-Wesley, Reading, MA, p. 478 and p. 571, 1990.
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LINKS
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Table of n, a(n) for n=0..107.
Eric Weisstein, Fibonorial Mathworld.
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FORMULA
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C2 = product((1-alpha^(2*k)), k>=1)) with alpha = (-1/phi^2) and phi = (1+sqrt(5))/2.
C2 = sum((-1)^binomial(n+1,2)*alpha^A152749(n), n>=0)
C2 = A062073/A194160
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EXAMPLE
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C2 = 0.83288324403391298245025664…
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MATHEMATICA
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digits = 108; NProduct[1 - GoldenRatio^(-4*k), {k, 1, Infinity}, WorkingPrecision -> digits+10, NProductFactors -> 200] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 14 2013, from 1st formula *)
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CROSSREFS
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Cf. A003266 and A062073; A194158 and A194160; A194157 and A194159.
Sequence in context: A063568 A198494 A100668 * A154158 A100863 A021986
Adjacent sequences: A194156 A194157 A194158 * A194160 A194161 A194162
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KEYWORD
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nonn,cons,easy
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AUTHOR
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Johannes W. Meijer, Aug 21 2011
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STATUS
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approved
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