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A194159
Constant associated with the product of the first n nonzero even-indexed Fibonacci numbers.
7
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
OFFSET
0,1
COMMENTS
a(n) = Product_{i=1..n} F(2*i) 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
Eric Weisstein, Fibonorial Mathworld.
FORMULA
Equals Product_{k>=1} (1-alpha^(2*k)) with alpha = -1/phi^2 and phi = (1+sqrt(5))/2.
Equals Sum_{n>=0} (-1)^binomial(n+1,2)*alpha^A152749(n).
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. A349272.
Sequence in context: A302138 A198494 A100668 * A154158 A100863 A021986
KEYWORD
nonn,cons,easy,changed
AUTHOR
Johannes W. Meijer, Aug 21 2011
STATUS
approved