A194160 Constant associated with the product of the first n nonzero odd-indexed Fibonacci numbers.

1, 4, 7, 2, 8, 8, 5, 9, 2, 9, 0, 9, 9, 5, 6, 9, 3, 1, 4, 6, 0, 7, 1, 3, 4, 2, 8, 1, 5, 0, 3, 8, 1, 5, 9, 3, 2, 2, 6, 9, 6, 2, 9, 5, 1, 5, 2, 6, 5, 6, 9, 9, 0, 5, 3, 7, 1, 1, 1, 5, 8, 6, 2, 3, 7, 6, 2, 7, 3, 6, 4, 9, 2, 8, 7, 7, 0, 5, 3, 7, 4, 4, 8, 2, 0, 5, 3, 1, 5, 9, 0, 6, 0, 9, 3, 6, 0

Offset: 1

Author

Johannes W. Meijer, Aug 21 2011

Keywords

• nonn
• easy
• cons

Comments

A194158(n) = prod(i=1..n, F(2*i-1) ) is asymptotic to C1*phi^(n*n)/sqrt(5)^n where phi=(1+sqrt(5))/2 and F(n) = A000045(n). The decimal expansion of the constant C1 is given here.

Examples

			C1 = 1.4728859290995693146071...
		

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.

Crossrefs

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

Programs

Mathematica

RealDigits[Product[1-((-1)/GoldenRatio^2)^(2k-1),{k,1000}],10, 100] [[1]] (* Harvey P. Dale, Aug 28 2011 *)
RealDigits[QPochhammer[-GoldenRatio^2, 1/GoldenRatio^4]/(GoldenRatio Sqrt[5]), 10, 100][[1]] (* Vladimir Reshetnikov, Sep 15 2016 *)

Formula

C1 = prod(k>=1, 1-alpha^(2*k-1) ) where alpha = (-1/phi^2) and phi = (1+sqrt(5))/2.

C1 = A062073/A194159