sqrt_2_approximation.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# 01 December 2016

# Approximation for sqrt(2).

# Numerator
#   a(n) = cos(2^(n-1) * arccos(3))

# Denominator
#   b(1) = 1
#   b(2) = 2
#   b(n) = 2 * a(n-2) * b(n-1)

# Fun facts:
#   gcd(a(n), b(n+1)) = 1
#   cos(acos(3) / 2) = sqrt(2)

# See also:
#   a(n) = https://oeis.org/A001601
#   b(n) = https://oeis.org/A051009

# Numerator
func a(n) {
    cos(2**(n-1) * acos(3))
}

# Denominator
func b((1)) { 1 }
func b((2)) { 2 }

func b(n) is cached {
    2 * b(n-1) * a(n-2)
}

for n in (1..10) {
    say a(n)/b(n+1)
}