A349272 -id:A349272 - cp's OEIS frontend

A194159 Constant associated with the product of the first n nonzero even-indexed Fibonacci numbers.

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

Johannes W. Meijer, Aug 21 2011

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.

			C2 = 0.83288324403391298245025664...

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.

Eric Weisstein, Fibonorial Mathworld.

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

Cf. A349272.

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 *)

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).
Equals A062073/A194160.

cp's OEIS Frontend

A194159 Constant associated with the product of the first n nonzero even-indexed Fibonacci numbers.

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