Pentagonal number

A visual representation of the first six pentagonal numbers

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 nth 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:

$p_{n}={\tfrac {3n^{2}-n}{2}}$

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 (sequence A000326 in the OEIS).

The nth pentagonal number is one third of the 3n-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, 1080, 1107, 1162, 1190, 1247, 1276, 1335... (sequence A001318 in the OEIS).

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.

Generalized pentagonal numbers and centered hexagonal numbers

Generalized pentagonal numbers are closely related to centered hexagonal numbers. When the array corresponding to a centered hexagonal number is divided between its middle row and an adjacent row, it appears as the sum of two generalized pentagonal numbers, with the larger piece being a pentagonal number proper:

1=1+0		7=5+2		19=12+7		37=22+15

In general:

$3n(n-1)+1={\tfrac {1}{2}}n(3n-1)+{\tfrac {1}{2}}(1-n)(3(1-n)-1)$

where both terms on the right are generalized pentagonal numbers and the first term is a pentagonal number proper (n ≥ 1). This division of centered hexagonal arrays gives generalized pentagonal numbers as trapezoidal arrays, which may be interpreted as Ferrers diagrams for their partition. In this way they can be used to prove the pentagonal number theorem referenced above.

Tests for pentagonal numbers

The simplest way to test whether a positive integer x is a (non-generalized) pentagonal number is by computing

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

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

The perfect square test

For generalized pentagonal numbers, it is sufficient to just check if

$24x+1$ is a perfect square.

For non-generalized pentagonal numbers, in addition to the perfect square test, it is also required to check if

${\sqrt {24x+1}}=5\mod 6$

The mathematical properties of Pentagonal numbers ensure that these tests are sufficient for proving or disproving the pentagonality of a number.

See also

• Hexagonal number
• Triangular number

Further Reading

• Leonard Euler: On the remarkable properties of the pentagonal numbers