A062073 revision #58
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A062073
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Decimal expansion of Fibonacci factorial constant.
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33
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1, 2, 2, 6, 7, 4, 2, 0, 1, 0, 7, 2, 0, 3, 5, 3, 2, 4, 4, 4, 1, 7, 6, 3, 0, 2, 3, 0, 4, 5, 5, 3, 6, 1, 6, 5, 5, 8, 7, 1, 4, 0, 9, 6, 9, 0, 4, 4, 0, 2, 5, 0, 4, 1, 9, 6, 4, 3, 2, 9, 7, 3, 0, 1, 2, 1, 4, 0, 2, 2, 1, 3, 8, 3, 1, 5, 3, 1, 2, 1, 6, 8, 4, 5, 2, 6, 2, 1, 5, 6, 2, 4, 9, 4, 7, 9, 7, 7, 4, 1, 2, 5, 9, 1, 3
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OFFSET
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1,2
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COMMENTS
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The Fibonacci factorial constant is associated with the Fibonacci factorial A003266.
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.5.
R. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison Wesley, 1990, pp. 478, 571.
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LINKS
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FORMULA
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C = (1-a)*(1-a^2)*(1-a^3)... 1.2267420... where a = -1/phi^2 and where phi is the Golden ratio = 1/2 + sqrt(5)/2.
C = exp( Sum_{k>=1} 1/(k*(1-(-(3+sqrt(5))/2)^k)) ). - Vaclav Kotesovec, Jun 08 2013
C = Sum_{k = -inf .. inf} (-1)^((k-1)*k/2) / phi^((3*k-1)*k), where phi = (1 + sqrt(5))/2. - Vladimir Reshetnikov, Sep 20 2016
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EXAMPLE
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1.226742010720353244417630230455361655871409690440250419643297301214...
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MATHEMATICA
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RealDigits[N[QPochhammer[-1/GoldenRatio^2], 105]][[1]] (* Alonso del Arte, Dec 20 2010 *)
RealDigits[N[Re[(-1)^(1/24) * GoldenRatio^(1/12) / 2^(1/3) * EllipticThetaPrime[1, 0, -I/GoldenRatio]^(1/3)], 120]][[1]] (* Vaclav Kotesovec, Jul 19 2015, after Eric W. Weisstein *)
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PROG
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(PARI) \p 1300 a=-1/(1/2+sqrt(5)/2)^2; prod(n=1, 17000, (1-a^n))
(PARI) { default(realprecision, 5080); p=-1/(1/2 + sqrt(5)/2)^2; x=prodinf(k=1, 1-p^k); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b062073.txt", n, " ", d)) } \\ Harry J. Smith, Jul 31 2009
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CROSSREFS
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KEYWORD
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AUTHOR
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Jason Earls (zevi_35711(AT)yahoo.com), Jun 27 2001
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STATUS
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approved
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