sum_of_squares_function_recursive.sf

#!/usr/bin/ruby

# Author: Daniel "Trizen" Șuteu
# Date: 12 August 2021
# https://github.com/trizen

# Count the number of ways or representing n as a sum of k squares (function known as: r_k(n)).

# See also:
#   https://en.wikipedia.org/wiki/Sum_of_squares_function

# OEIS sequences:
#   https://oeis.org/A004018 -- Theta series of square lattice (or number of ways of writing n as a sum of 2 squares). Often denoted by r(n) or r_2(n).
#   https://oeis.org/A005875 -- Theta series of simple cubic lattice; also number of ways of writing a nonnegative integer n as a sum of 3 squares (zero being allowed).
#   https://oeis.org/A000118 -- Number of ways of writing n as a sum of 4 squares; also theta series of lattice Z^4.
#   https://oeis.org/A000132 -- Number of ways of writing n as a sum of 5 squares.
#   https://oeis.org/A000141 -- Number of ways of writing n as a sum of 6 squares.
#   https://oeis.org/A008451 -- Number of ways of writing n as a sum of 7 squares.

func r(n, k=2) is cached {

    return 1 if (n == 0)
    return 0 if (k <= 0)

    return (n.is_square ? 2 : 0) if (k == 1)

    var count = 0

    for a in (0 ..  n.isqrt) {
        if (k > 2) {
            count += (a.is_zero ? 1 : 2)*__FUNC__(n - a**2, k-1)
        }
        elsif (n - a**2 -> is_square) {
            count += (a.is_zero ? 1 : 2)*(n - a**2 -> is_zero ? 1 : 2)
        }
    }

    return count
}

for k in (0..9) {
    say ("k = #{k}: ", 15.of { r(_, k) })
}

__END__
k = 0: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
k = 1: [1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0]
k = 2: [1, 4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0]
k = 3: [1, 6, 12, 8, 6, 24, 24, 0, 12, 30, 24, 24, 8, 24, 48]
k = 4: [1, 8, 24, 32, 24, 48, 96, 64, 24, 104, 144, 96, 96, 112, 192]
k = 5: [1, 10, 40, 80, 90, 112, 240, 320, 200, 250, 560, 560, 400, 560, 800]
k = 6: [1, 12, 60, 160, 252, 312, 544, 960, 1020, 876, 1560, 2400, 2080, 2040, 3264]
k = 7: [1, 14, 84, 280, 574, 840, 1288, 2368, 3444, 3542, 4424, 7560, 9240, 8456, 11088]
k = 8: [1, 16, 112, 448, 1136, 2016, 3136, 5504, 9328, 12112, 14112, 21312, 31808, 35168, 38528]
k = 9: [1, 18, 144, 672, 2034, 4320, 7392, 12672, 22608, 34802, 44640, 60768, 93984, 125280, 141120]