project-euler/Sidef/140 Modified Fibonacci golden nuggets.sf

#!/usr/bin/ruby

# Author: Daniel "Trizen" Șuteu
# License: GPLv3
# Website: https://github.com/trizen

# https://projecteuler.net/problem=140

# Runtime: 0.149s

# After some analysis, I ended up with the following formula:
#
#      f(x) = -(x(3x + 1)) / (x^2 + x - 1)
#
# which generates golden nuggets when it's called by the following rules:
#
# when i is even: f(fib(i) / fib(i+1))
# when i is odd:  f((fib(i+1)+lucas(i)) / (fib(i+2)+lucas(i+1)))

func lucas(n) is cached {
    n.is_one ? 1
             : (n.is_zero ? 2
                          : (lucas(n-1) + lucas(n-2)))
}

func fib(n) is cached {
    n.is_one ? 1
             : (n.is_zero ? 0
                          : (fib(n-1) + fib(n-2)))
}

func f(x) {
    - x*(3*x + 1) / (x**2 + x - 1) -> round
}

var sum = 0

for i in (1..30) {
    if (i.is_even) {
        sum += f(fib(i)/fib(i+1))
    }
    else {
        var a = fib(i+1)+lucas(i)
        var b = fib(i+2)+lucas(i+1)
        sum += f(a/b)
    }
}

say sum