Centered triangular number

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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 ${\displaystyle n}$.

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).