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A194159 Constant associated with the product of the first n nonzero even-indexed Fibonacci numbers. 6
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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.

REFERENCES

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.

LINKS

Table of n, a(n) for n=0..107.

Eric Weisstein, Fibonorial Mathworld.

FORMULA

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

EXAMPLE

C2 = 0.83288324403391298245025664…

MATHEMATICA

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 *)

RealDigits[QPochhammer[1/GoldenRatio^4], 10, 100][[1]] (* Vladimir Reshetnikov, Sep 15 2016 *)

CROSSREFS

Cf. A003266 and A062073; A194158 and A194160; A194157 and A194159.

Sequence in context: A302138 A198494 A100668 * A154158 A100863 A021986

Adjacent sequences:  A194156 A194157 A194158 * A194160 A194161 A194162

KEYWORD

nonn,cons,easy

AUTHOR

Johannes W. Meijer, Aug 21 2011

STATUS

approved

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Last modified January 19 06:37 EST 2020. Contains 331033 sequences. (Running on oeis4.)