A centered (or centred) triangular number is a centeredfigurate number that represents an equilateral triangle with a dot in the center and all its other dots surrounding the center in successive equilateral triangular layers.

This is also the number of points of a hexagonal lattice with nearest-neighbor coupling whose distance from a given point
is less than or equal to $n$.

The following image shows the building of the centered triangular numbers by using the associated figures: at each step, the previous triangle (shown in red) is surrounded by a triangular layer of new dots (in blue).

Properties

The gnomon of the n-th centered triangular number, corresponding to the (n + 1)-th triangular layer, is:

$C_{3,n+1}-C_{3,n}=3(n+1).$

The n-th centered triangular number, corresponding to n layers plus the center, is given by the formula:

If the centered triangular numbers are treated as the coefficients of
the McLaurin series of a function, that function converges for all $|x|<1$, in which case it can be expressed as the meromorphic generating function