Centered triangular number

A centered (or centred) triangular number is a centered figurate 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

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).

The centered triangular numbers can be expressed in terms of the centered square numbers: where The first centered triangular numbers (C3,n < 3000) are: The first simultaneously triangular and centered triangular numbers (C3,n = TN < 109) are: If the centered triangular numbers are treated as the coefficients of the McLaurin series of a function, that function converges for all

, in which case it can be expressed as the meromorphic generating function