Polygonal Number
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a polygonal number is a
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
represented as dots or pebbles arranged in the shape of a
regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
. The dots are thought of as alphas (units). These are one type of 2-dimensional
figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polygon ...
s.


Definition and examples

The number 10 for example, can be arranged as a
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
(see
triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
): : But 10 cannot be arranged as a
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
. The number 9, on the other hand, can be (see
square number In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
): : Some numbers, like 36, can be arranged both as a square and as a triangle (see
square triangular number In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are infinitely many square triangular numbers; the first few are: :0, 1, 36, , , , , , , Expl ...
): : By convention, 1 is the first polygonal number for any number of sides. The rule for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between those points. In the following diagrams, each extra layer is shown as in red.


Triangular numbers

:


Square numbers

: Polygons with higher numbers of sides, such as pentagons and hexagons, can also be constructed according to this rule, although the dots will no longer form a perfectly regular lattice like above.


Pentagonal numbers

:


Hexagonal numbers

:


Formula

If is the number of sides in a polygon, the formula for the th -gonal number is :P(s,n) = \frac or :P(s,n) = (s-2)\frac+n The th -gonal number is also related to the triangular numbers as follows: :P(s,n) = (s-2)T_ + n = (s-3)T_ + T_n\, . Thus: :\begin P(s,n+1)-P(s,n) &= (s-2)n + 1\, ,\\ P(s+1,n) - P(s,n) &= T_ = \frac\, ,\\ P(s+k,n) - P(s,n) &= k T_ = k\frac\, . \end For a given -gonal number , one can find by :n = \frac and one can find by :s = 2+\frac\cdot\frac.


Every hexagonal number is also a triangular number

Applying the formula above: :P(s,n) = (s-2)T_ + n to the case of 6 sides gives: :P(6,n) = 4T_ + n but since: :T_ = \frac it follows that: :P(6,n) = \frac + n = \frac = T_ This shows that the th hexagonal number is also the th triangular number . We can find every hexagonal number by simply taking the odd-numbered triangular numbers: :1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...


Table of values

The first 6 values in the column "sum of reciprocals", for triangular to octagonal numbers, come from a published solution to the general problem, which also gives a general formula for any number of sides, in terms of the
digamma function In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac. It is the first of the polygamma functions. It is strictly increasing and strict ...
. The
On-Line Encyclopedia of Integer Sequences The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
eschews terms using Greek prefixes (e.g., "octagonal") in favor of terms using numerals (i.e., "8-gonal"). A property of this table can be expressed by the following identity (see ): :2\,P(s,n) = P(s+k,n) + P(s-k,n), with :k = 0, 1, 2, 3, ..., s-3.


Combinations

Some numbers, such as 36 which is both square and triangular, fall into two polygonal sets. The problem of determining, given two such sets, all numbers that belong to both can be solved by reducing the problem to
Pell's equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinate ...
. The simplest example of this is the sequence of
square triangular number In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are infinitely many square triangular numbers; the first few are: :0, 1, 36, , , , , , , Expl ...
s. The following table summarizes the set of -gonal -gonal numbers for small values of and . : In some cases, such as and , there are no numbers in both sets other than 1. The problem of finding numbers that belong to three polygonal sets is more difficult. A computer search for pentagonal square triangular numbers has yielded only the trivial value of 1, though a proof that there are no other such numbers has yet to be found. The number 1225 is hecatonicositetragonal (), hexacontagonal (), icosienneagonal (), hexagonal, square, and triangular. The only polygonal set that is contained entirely in another polygonal set is the set of hexagonal numbers, which is contained in the set of triangular numbers.


See also

*
Centered polygonal number The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides. Each side of a polygonal layer contains one more dot than each side i ...
*
Polyhedral number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polygon ...
*
Fermat polygonal number theorem In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most -gonal numbers. That is, every positive integer can be written as the sum of three or fewer triangular numbers, and as the sum ...
*


Notes


References

*''
The Penguin Dictionary of Curious and Interesting Numbers ''The Penguin Dictionary of Curious and Interesting Numbers'' is a reference book for recreational mathematics and elementary number theory written by David Wells. The first edition was published in paperback by Penguin Books in 1986 in the UK, ...
'',
David Wells David Lee Wells (born May 20, 1963) is an American former baseball pitcher who played 21 seasons in Major League Baseball (MLB) for nine teams, most notably the Toronto Blue Jays and New York Yankees. Nicknamed "Boomer", Wells was considered on ...
(
Penguin Books Penguin Books is a British publishing, publishing house. It was co-founded in 1935 by Allen Lane with his brothers Richard and John, as a line of the publishers The Bodley Head, only becoming a separate company the following year.Polygonal numbers at PlanetMath
* *


External links

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* * Polygonal Number Counting Function: http://www.mathisfunforum.com/viewtopic.php?id=17853 {{Authority control Figurate numbers