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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... A figurate number is a number that can be represented as a regular and discrete geometric pattern (e. ... A triangular number (so called because it can be arranged into a triangle) is the sum of the n natural numbers from 1 to n. ... In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer. ... Look up pentagon in Wiktionary, the free dictionary. ... The triskelion appearing on the Isle of Man flag. ... In geometry, a vertex (plural vertices) is a special kind of point, usually a corner of a polygon, polyhedron, or higher dimensional polytope. ...

p_n is given by the formula:

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 OEIS) Look up one in Wiktionary, the free dictionary. ... Look up five in Wiktionary, the free dictionary. ... Look up twelve in Wiktionary, the free dictionary. ... 22 (twenty-two) is the natural number following 21 and preceding 23. ... 35 (thirty-five) is the natural number following 34 and preceding 36. ... 51 (fifty-one) is the natural number 51 following 50 and preceding 52. ... Look up seventy in Wiktionary, the free dictionary. ... 92 is the natural number following 91 and preceding 93. ... 117 is the natural number following 116 and preceding 118. ... 1001 is the natural number preceding 1002 and following 1000. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

The nth pentagonal number is one third of the 3n-1th triangular number. A triangular number (so called because it can be arranged into a triangle) is the sum of the n natural numbers from 1 to n. ...

Pentagonal numbers are important to Euler's theory of partitions, as expressed in his pentagonal number theorem. Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... In number theory, a partition of a positive integer n is a way of writing n as a sum of positive integers. ... In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. ...

"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...

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. A centered pentagonal number is a centered figurate number that represents a pentagon with a dot in the center and all other dots surrounding the center in successive pentagonal layers. ...