A **decagonal number** is a figurate number that extends the concept of triangular and square numbers to the decagon (a ten-sided polygon). However, unlike the triangular and squjare numbers, the patterns involved in the construction of decagonal numbers are not rotationally symmetrical. Specifically, the *n*th decagonal numbers counts the number of dots in a pattern of *n* nested decagons, all sharing a common corner, where the *i*th decagon in the pattern has sides made of *i* dots spaced one unit apart from each other. The *n*-th decagonal number is given by the formula

- $ D_n = 4n^2 - 3n. $

The first few decagonal numbers are:

- 0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, 4000, 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326...

The *n*-th decagonal number can also be calculated by adding the square of *n* to thrice the (*n*—1)-th pronic number or, to put it algebraically, as

- $ D_n = n^2 + 3(n^2 - n). $

## Properties Edit

- Decagonal numbers consistently alternate parity.

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