On to the heptagonal

I couldn't help keep thinking about the polygon "figurate" number patterns on the drive to work this morning.  After the hexagon comes the seven-sided heptagon.

But first lets review the pattern.

The triangular number series is generated by n(n-1)/2.

The square numbers are generated by n^2, but also as the sum of two consecutive triangular numbers n(n-1)/2 + n(n+1)/2, just as a square can geometrically be divided into two triangles. 

The pentagonal numbers are sums of two consecutive triangular numbers, except with two of the smaller number:  2(n(n-1)/2) + n(n+1)/2

The hexagonal numbers of sums of a square number of two triangular numbers, and the square component can be reduced to two triangulars.  This comes out to: 3(n(n-1)/2) + n(n+1)/2

The pattern emerging is, the figurate number series corresponding to an s sided polygon is (s-3)(n(n-1)/2) + n(n+1)/2

If this is true than the heptagonal numbers should be 4(n(n-1)/2) + n(n+1)/2.  This gives 1, 7, 18, 34, 55, ... and these are the heptagonal numbers!

Also, two of the triangles can be combined to a square number which gives:

 (s-4)(n(n-1)/2) + n^2

-----

At this point I did some more searching online; this seems so obvious it must be out there.  In fact there is a wikipedia page devoted to polygonial numbers that I missed before (I had been searching too specifically for "pentagonal numbers," "triangular numbers" or "figurate number").   Under formulae it says:

The n^th s-gonal number is also related to the triangular numbers T_n as follows:

which is equivalent to the general case I came up with above.

(s-3)(n(n-1)/2) + n(n+1)/2