It is not. There are indeed exactly 8 ways to pick three elements from the set [1,9] such that their sum equals 15. But for 4x4, there are 86 (!) ways to pick four elements from [1, 15] so that their sum is 34 (the magic constant for 4x4), whereas only ten of the combinations are used in construction of the square.

Some Python:

  from collections import Counter
  from itertools import combinations

  def number_of_sums(n, M):
    combs = combinations(range(1,n**2+1), n)
    sums = (sum(c) for c in combs)
    return Counter(sums)[M]

  print(number_of_sums(3, 15))
  print(number_of_sums(4, 34))