A194159 - OEIS

%I #20 Oct 22 2024 05:23:49

%S 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,

%T 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,

%U 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

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

%C 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.

%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="http://mathworld.wolfram.com/Fibonorial.html">Fibonorial</a> Mathworld.

%F Equals Product_{k>=1} (1-alpha^(2*k)) with alpha = -1/phi^2 and phi = (1+sqrt(5))/2.

%F Equals Sum_{n>=0} (-1)^binomial(n+1,2)*alpha^A152749(n).

%F Equals A062073/A194160.

%e C2 = 0.83288324403391298245025664...

%t 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 *)

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

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

%Y Cf. A349272.

%K nonn,cons,easy

%O 0,1

%A _Johannes W. Meijer_, Aug 21 2011