Proving sequence formula - stellar numbers problem

Hi,

I've been working on a project based upon the idea of 'stellar numbers'. That is, a number which, when represented as dots, can be arranged into the shape of a star with *p* vertices. This leads to a sequence for each value of *p*, for example when *p* is 6 (i.e. a six pointed star) the first five terms (S_{1} to S_{5}) would be:

*1, 13, 37, 73, 121....*

and when is 3:

*1, 7, 19, 37, 61...*

etc.

I worked out the first eight terms of the sequences for five values of *p* and found a general statement: *S*_{n }= pn^{2} - pn + 1

This obviously works for all terms I have actually found however I would like to know if there is a way that I can mathematically prove that this will work for any value of *p* or *n* (providing they're natural numbers of course). Or is this unnecessary? Impossible?

Any help would be appreciated :)

Thanks,

jdbt

Re: Proving sequence formula - stellar numbers problem

you could try a proof by Mathematical induction