perfect_squares_in_catalan_s_triangle.sf

#!/usr/bin/ruby

# Perfect squares in Catalan's triangle, in column 2.

# Equivalent with finding integers n, such that:
#   binomial(n+2, 2) - binomial(n+2, 1) = k^2

# Also equivalent with:
#   (n + 1)*(n + 2)/2 - 2 - n = k^2

# See also:
#   https://oeis.org/A317027

# This provides upper bounds in A317027:
#   A317027(17) <= 26039560591564569275625
#   A317027(18) <= 884578527018516731261184
#   A317027(19) <= 30049630358038004293604649
#   A317027(20) <= 1020802853646273629251296900

func a(m) {
    (3*(3 - sqrt(8))**m + 3*(3 + sqrt(8))**m - 2)/4
}

for m in (1..20) {
    var n = a(m)
    say ("C(#{n}, 2) = ", (n + 1)*(n + 2)/2 - 2 - n)
}