centred hexagonal numbers and bridge cables

a ppt is here

a quadratic generalisation

links with the difference between consecutive cubes
and this is no coincidence (see here)

this general form is also
n^3 - (n - 1)^3

summing the terms always gives a cube number

(e.g. 1 + 7 + 19 + 37 = 64 = 4^3)

 n = 4

 n = 5

 n = 6

n = 8


















centred hexagonal numbers are closely related to triangular numbers: