Editing Decagonal number (section)

{{Short description|Figurate number representing a decagon}}
In [[mathematics]], a '''decagonal number''' is a [[figurate number]] that extends the concept of [[triangular number|triangular]] and [[square number]]s to the [[decagon]] (a ten-sided polygon). However, unlike the triangular and square numbers, the patterns involved in the construction of decagonal numbers are not rotationally symmetrical. Specifically, the ''n''-th decagonal numbers counts the 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 following formula
: <math>d_n = 4n^2 - 3n</math>.
The first few decagonal numbers are:
: [[0 (number)|0]], [[1 (number)|1]], [[10 (number)|10]], [[27 (number)|27]], [[52 (number)|52]], [[85 (number)|85]], [[126 (number)|126]], [[175 (number)|175]], [[232 (number)|232]], [[297 (number)|297]], 370, 451, 540, 637, 742, 855, 976, [[1105 (number)|1105]], 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, [[4000 (number)|4000]], 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326 {{OEIS|id=A001107}}.

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 
: <math>D_n = n^2 + 3\left(n^2 - n\right)</math>.