number_of_representations_as_sum_of_3_triangles.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 02 March 2018
# https://github.com/trizen

# Compute the number of ordered ways of writing `n` as the sum of 3 triangular numbers.

# See also:
#   https://oeis.org/A008443
#   https://projecteuler.net/problem=621

func count_sums_of_two_squares (n) {

    var count = 1

    for p,e in (n.factor_exp) {

        var r = p%4

        if (r == 3) {
            e.is_even || return 0
        }

        if (r == 1) {
            count *= (e + 1)
        }
    }

    return count
}

func count_triangular_sums (n) {
    0..n.ipolygonal_root(3) -> sum_by {|u|
        var z = (8*(n - (u * (u + 1) / 2)) + 1)
        count_sums_of_two_squares(z + 1)
    }
}

say count_triangular_sums(1000)     #=> 78
say count_triangular_sums(1234)     #=> 144
say count_triangular_sums(10**6)    #=> 2106