Pentagonal and Triagonal Numbers

The last post got me to thinking.  If a square can be divided into two triangles, and a square number is the sum of two triangular numbers, is a pentagonal number the sum of one larger and two smaller triangular numbers?

The first pentagonal numbers are 1, 5, 12, 22, 35, ...  The triangular numbers are 1, 3, 6, 10, 15, ...  Sure enough, 5 = 3 + 1 +1, 12 = 6 + 3 + 3, 22 = 10 + 6 + 6, 35 = 15 + 10 + 10...  The series of pentagonal numbers are made up of consecutive triangular numbers, two of the smaller one and one of the larger. 

...then can a hexagonal number be reduced to a rectangular number and two equal triangular numbers?

The hexagonal numbers are 1, 6, 15, 28, 45, ...  And indeed 6 = 4 + 1 + 1, 15 = 9 + 3 +3, 28 = 16 + 6 + 6, 45 = 25 + 10 + 10, ...  The series of hexagonal numbers is equal to the sums two triangular numbers and a square number (a special case of a rectangular number).

From this point there are obvious ways to reduce the hexagon and further polygons.  Perhaps this connection to number series is obvious to a mathematician, and is well known in mathematics, but I have not heard of it. It is fun to come up with something new (at least new to me) that appears to work.