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 (S1 to S5) would be:
1, 13, 37, 73, 121....
and when is 3:
1, 7, 19, 37, 61...
I worked out the first eight terms of the sequences for five values of p and found a general statement: Sn = pn2 - 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