A194160 - OEIS

%I #16 Feb 16 2025 08:33:15

%S 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,

%T 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,

%U 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

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

%C 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.

%D 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.

%H Eric Weisstein, <a href="https://mathworld.wolfram.com/Fibonorial.html">Fibonorial</a>, Mathworld.

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

%F C1 = A062073/A194159

%e C1 = 1.4728859290995693146071...

%t RealDigits[Product[1-((-1)/GoldenRatio^2)^(2k-1),{k,1000}],10, 100] [[1]] (* _Harvey P. Dale_, Aug 28 2011 *)

%t RealDigits[QPochhammer[-GoldenRatio^2, 1/GoldenRatio^4]/(GoldenRatio Sqrt[5]), 10, 100][[1]] (* _Vladimir Reshetnikov_, Sep 15 2016 *)

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

%K nonn,easy,cons

%O 1,2

%A _Johannes W. Meijer_, Aug 21 2011