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. A figurate number is a number that can be represented as a regular and discrete geometric pattern (e A triangular number 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 Regular pentagons The term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation. 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. In Geometry, a vertex (plural "vertices" is a special kind of point. 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:

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)

The nth pentagonal number is one third of the 3n-1th triangular number. Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity This article discusses the number five. For the year 5 AD see 5. 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. 70 ( seventy) is the Natural number following 69 and preceding 71. 92 ( ninety-two) is the Natural number following 91 and preceding 93. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences A triangular number is the sum of the n Natural numbers from 1 to n.

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. . . (sequence A001318 in OEIS)

Generalized pentagonal numbers are important to Euler's theory of partitions, as expressed in his pentagonal number theorem. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences 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.

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

Tests for pentagonal numbers

One can test whether a positive integer x is a pentagonal number by computing

If n is an integer, then x is the nth pentagonal number. If n is not an integer, then x is not pentagonal.

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