The values of s are: 4, 8, 9, 10, 12, 14, 16, 18, 22, 24, 32.
It can be seen that n is, on average, an increasing function. (It is constant at s = 8 and s = 9 and decreases at s = 12). If proved this would show there is no repetition of a value of n for which simultaneously s! + n^2 = b^2 and (s+k)! + n^2 = c^2 for general and large values of k (not only for k = 1) and would solve Brocard's Problem: Exactly, the only 3 solutions to s! + 1 = b^2 are (4,5); (5,11) and (7,71).
Note that n^2 was chosen a square, but this is not necessary.
More terms of the sequence are hard to get if the program based on a simple algorithm, needing 10^9 bytes memory, is not improved in the sense of reducing the number of divisors used. This could probably be done.
