A276499 Decimal expansion of Fibonorial(1/2).

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

Offset: 0

Author

Vladimir Reshetnikov, Sep 05 2016

Keywords

• nonn
• cons

Comments

This constant can be thought of as A003266(1/2).

The fibonorial of (not necessarily an integer) x is defined as x!F = (phi^(x*(x+1)/2) / F(x+1)) * Product{n=1..inf} F(n+1)^(x+1)/(F(n)^x * F(x+n+1)), where F(x) = (phi^x - cos(Pi*x)/phi^x)/sqrt(5), where phi = (1+sqrt(5))/2 is the golden ratio. It satisfies the recurrence 0!_F = 1, x!_F = (x-1)!_F * F(x), and agrees with A003266(x) at integer points.

Examples

			0.98260982501326431122377480560574910946538...
		

Links

• Simon Plouffe, Fibonacci factorials.
• Eric Weisstein's World of Mathematics, Fibonorial, Fibonacci Factorial Constant.

Crossrefs

Cf. A003266, A062073.

Programs

Mathematica

RealDigits[N[GoldenRatio^(3/8) QPochhammer[-1/GoldenRatio^2]/5^(1/4), 100]][[1]]

Formula

Fibonorial(1/2) = phi^(3/8) * C / 5^(1/4), where C = A062073 is the Fibonacci factorial constant.