Pentagonal number

A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The "n"th pentagonal number "p"_n is the number of "distinct" dots in a pattern of dots consisting of the "outlines" of regular pentagons whose sides contain 1 to n dots, overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside...

"p"_n is given by the formula:

:$p\_n\; =\; frac\{n(3n-1)\}2$

for "n" ≥ 1. The first few pentagonal numbers are:

1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001 OEIS|id=A000326.

The "n"th pentagonal number is one third of the 3"n"-1th triangular number.

Generalized pentagonal numbers are obtained from the formula given above, but with "n" taking values in the sequence 0, 1, -1, 2, -2, 3, -3, 4..., producing the sequence:

0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027... OEIS|id=A001318.

Generalized pentagonal numbers are important to Euler's theory of partitions, as expressed in his pentagonal number theorem.

The number of dots inside the outermost pentagon of a pattern forming a pentagonal number is itself a generalized pentagonal number.

Pentagonal numbers should not be confused with centered pentagonal numbers.

Tests for pentagonal numbers

One can test whether a positive integer "x" is a (non-generalized) pentagonal number by computing

:$n\; =\; frac\{sqrt\{24x+1\}\; +\; 1\}\{6\}.$

If "n" is a natural number, then "x" is the "n"th pentagonal number. If "n" is not a natural number, then "x" is not pentagonal.