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 Different Sums (Posted on 2004-07-22)
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?

 See The Solution Submitted by Brian Smith Rating: 3.2500 (4 votes)

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 Curious second differences | Comment 8 of 17 |
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?
 Posted by Federico Kereki on 2004-07-22 11:09:54

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