I can prove that it's not a function x^k where k is a natural number <=7 by taking the numbers and figuring out the difference between the numbers in the sequence, then figuring out the differences between those.

The first 10 numbers (and differences) are:

0 80 135 225 375 620 1040 1735 2900 4840 8080 1 55 90 150 245 420 695 1165 1940 3240 2 35 60 95 175 275 470 775 1300 3 25 35 80 100 195 305 525 4 10 452095 110 220 5 35 -25 75 15 110 6 -55 100 -60 95 7 155 -160 155

It's worth noting the point of inflection on the 4th level. I don't have more than those 10 numbers unfortunately. It's tempting to believe that the next one on the 7th row would be -160 though.

The numbers always end in either 0 or 5, though they don't switch between 0 and 5 in any sort of regular pattern.

That's all I can think of off hand to try and crack that problem.

Incidentally, this is (in theory) part of a class of functions. Those numbers correspond to resource counts for a game (Travian incidentally) and I'm trying to predict how it grows for a program I'm writing.