There are 40 ways to make sums of three distinct positive integers total 25. (1+2+22 is such a sum, but 1+12+12 and 1+2+3+19 are not.)

How many different ways can three distinct positive integers sum to 1000?

(In reply to

Curious second differences by Federico Kereki)

If you just pick the numbers for 6, 12, 18, 24... and so on, you find that the answer is N²/12-N/2+1.