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

1,2

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.

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=1..97.

Eric Weisstein, Fibonorial, Mathworld.

Formula

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

C1 = A062073/A194159

Example

C1 = 1.4728859290995693146071...

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 *)

Crossrefs

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

Sequence in context: A021683 A248179 A362753 * A154466 A096316 A010777

Adjacent sequences: A194157 A194158 A194159 * A194161 A194162 A194163

Keyword

nonn,easy,cons

Author

Johannes W. Meijer, Aug 21 2011

Status

approved