viswanath_s_constant_conjecture.sf

#!/usr/bin/ruby

# Author: Daniel "Trizen" Șuteu
# Date: 24 February 2018
# https://github.com/trizen

# A new formula for computing Viswanath's constant. (weak conjecture -- incorrect)

# Blandin-Diaz compositional Bernoulli numbers (B^S)_1,n:
#  a(0) = 1
#  a(n) = -Sum_{k=0..n-1} a(k) / ((n-k+1)!)^2

# Next, we define `t` to be the following sum:
#   t = Sum_{k=0..Infinity} 1 / (a(k) * k!)

# From which we extract Viswanath's constant:
#   v = (19500 - 3*t)/(10756 - 3*t)

# See also:
#   https://oeis.org/A078416
#   https://en.wikipedia.org/wiki/Random_Fibonacci_sequence
#   https://www.wolframalpha.com/input/?i=Viswanath%27s+constant
#   https://www.wolframalpha.com/input/?i=18497.440650652720515613516713415750667794722534171

func a((0)) { 1 }

func a(n) is cached {
    -sum(^n, {|k| a(k) / (n - k + 1)!**2 })
}

var t = sum(0..200, {|k|
    1 / (a(k) * k!)
})

var v = (19500 - 3*t)/(10756 - 3*t)

say "Viswanath's constant is: #{v} (conjectured)"

__END__
Viswanath's constant is: 1.13198824879430090510505642117075909739291590469 (conjectured)