sidef-scripts/Math/unreduced_continued_fractions.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# License: GPLv3
# Date: 02 December 2016
# https://github.com/trizen

# Recursive evaluation of continued fractions rationally,
# by computing the numerator and the denominator individually.

# For every continued fraction, we have the following relation:
#
#    n
#   | / a(k)    kn(n)
#   |/ ----- = -------
#   | \ b(k)    kd(n)
#   k=0

#
## Example for:
##   K(n^2 / (2n+1)) = 4/Pi - 1
#

func a(n) { n**2 }
func b(n) { 2*n + 1 }

func kn(n) is cached {
    n < 2 ? (n == 0 ? 1 : 0)
          : (b(n - 1)*kn(n - 1) + a(n - 1)*kn(n - 2))
}

func kd(n) is cached {
    n < 2 ? n
          : (b(n - 1)*kd(n - 1) + a(n - 1)*kd(n - 2))
}

for n in (0 .. 10) {
    printf("%2d. %20s %-20s\n", n, kn(n), kd(n));
}