Figurate number

The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean

polygonal number
a number represented as a discrete $r$ -dimensional regular geometric pattern of $r$ -dimensional balls such as a polygonal number (for $r = 2$ ) or a polyhedral number (for $r = 3$ ).
a member of the subset of the sets above containing only triangular numbers, pyramidal numbers, and their analogs in other dimensions.

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