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?
Using a variant of my awk program, I found the answer to this problem for 6, 7, 8... and so on. When you calculate the second differences of the results you find a curious pattern (0,0,0,1,-1,1):
6 1
7 1 0
8 2 1 1
9 3 1 0
10 4 1 0
11 5 1 0
12 7 2 1
13 8 1 -1
14 10 2 1
15 12 2 0
16 14 2 0
17 16 2 0
18 19 3 1
19 21 2 -1
20 24 3 1
21 27 3 0
22 30 3 0
23 33 3 0
24 37 4 1
25 40 3 -1
26 44 4 1
27 48 4 0
28 52 4 0
29 56 4 0
30 61 5 1
31 65 4 -1
32 70 5 1
33 75 5 0
34 80 5 0
35 85 5 0
36 91 6 1
37 96 5 -1
38 102 6 1
39 108 6 0
40 114 6 0
41 120 6 0
42 127 7 1
43 133 6 -1
44 140 7 1
45 147 7 0
46 154 7 0
47 161 7 0
48 169 8 1
49 176 7 -1
50 184 8 1
51 192 8 0
52 200 8 0
53 208 8 0
54 217 9 1
This seems to suggest there is a simple (?) formula... which?
Curious second differences
