An octagonal number is a figurate number that represents an octagon. The octagonal number for n is given by the formula 3n2 - 2n, with n > 0. The first few octagonal numbers are:
1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936 (sequence A000567 in OEIS)
Octagonal numbers can be formed by placing triangular numbers on the four sides of a square. To put it algebraically, the nth octagonal number is
which simplifies to the formulas given above.
The octagonal number for n can also be calculated by adding the square of n to twice the (n - 1)th pronic number, or, to put it algebraically, .
Octagonal numbers consistently alternate parity.
Octagonal numbers are occasionally referred to as "star numbers," though that term is more commonly used to refer to centered dodecagonal numbers.
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“Hence, a generative grammar must be a system of rules that can iterate to generate an indefinitely large number of structures. This system of rules can be analyzed into the three major components of a generative grammar: the syntactic, phonological, and semantic components.”
—Noam Chomsky (b. 1928)