Consider the differences between powers of successive natural numbers. Note that the difference between two consecutive squares is always odd. Here is my picture of this relationship:

0 - 0
       - 1
1 - 1      - 2
       - 3
2 - 4      - 2
       - 5
3 - 9      - 2
       - 7
4 - 16     - 2
       - 9
5 - 25

Etc. But this sort of pattern is not limited to the squares. Notice the "last common difference" in the third powers of the integers:

0 - 0
        - 1
1 - 1        - 6
        - 7       - 6
2 - 8        - 12
        - 19      - 6
3 - 27       - 18
        - 37      - 6
4 - 64       - 24
        - 61      - 6
5 - 125      - 30
        - 91
6 - 216

If one continues this process for the fourth, fifth, etc. powers of numbers, the "last common difference" is the factorial of the power in question. Thus any power of a natural number is expressible as a simple sum. This process leads to the question: What about the 1.5 power? For an answer, see the gamma function, which is a close relative to the factorial.