Polygonal number

The number 10 for example, can be arranged as a triangle (see triangular number): : But 10 cannot be arranged as a square. The number 9, on the other hand, can be (see square number): : Some numbers, like 36, can be arranged both as a square and as a triangle (see square triangular number): : 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 : The triangular number sequence is the representation of the numbers in the form of equilateral triangle arranged in a series or sequence. These numbers are in a sequence of 1, 3, 6, 10, 15, 21, 28, 36, 45, and so on. 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 ==

Polygonal numbers differ mainly by the difference between each consecutive number. The difference between each triangular number increases by 1 each iteration. For square numbers the difference increases by 2. For pentagonal numbers it increases by 3. The size of the difference between each N starts at 1 for polygonal numbers (and for centered polygonal numbers the size of the difference starts at 0). For example: the first triangular is 1 and the "difference" starts at 1. To get the next triangular, you increment the difference by 1 and add it to the current one. The 2nd triangular number is 3, the difference is 2. Increment the difference again, it's 3. 3+3 is 6, the 3rd triangular. The difference between the 3rd and the 2nd is 3. Given that triangular numbers are the sum of the first N numbers this is unsurprising, however the same method holds for all other polygonal numbers as well. With squares for the Nth and diff we start at the first square 1 and a diff (or offset) of 1. For the next square the difference is first incremented by 2: 1+2=3. We then add the difference to the current square: 1+3=4. The 2nd square number. Continuing the pattern for Nth+diff: 4+5=9, 9+7=16, 16+9=25. In other words, for the Nth polygonal number the difference between it and the next is X * N + 1. Where X is the number of sides (beginning at 1 for triangular). If is the number of sides in a polygon, the formula for the th -gonal number is :P(s,n) = \frac{(s-2)n^2-(s-4)n}{2} The th -gonal number is also related to the triangular numbers as follows: :P(s,n) = (s-2)T_{n-1} + n = (s-3)T_{n-1} + T_n\, . Thus: :\begin{align} P(s,n+1)-P(s,n) &= (s-2)n + 1\, ,\\ P(s+1,n) - P(s,n) &= T_{n-1} = \frac{n(n-1)}{2}\, ,\\ P(s+k,n) - P(s,n) &= k T_{n-1} = k\frac{n(n-1)}{2}\, . \end{align} For a given -gonal number , one can find by :n = \frac{\sqrt{8(s-2)x+{(s-4)}^2}+(s-4)}{2(s-2)} and one can find by :s = 2+\frac{2}{n}\cdot\frac{x-n}{n-1}. Every hexagonal number is also a triangular number Applying the formula above: :P(s,n) = (s-2)T_{n-1} + n to the case of 6 sides gives: :P(6,n) = 4T_{n-1} + n but since: :T_{n-1} = \frac{n(n-1)}{2} it follows that: :P(6,n) = \frac{4n(n-1)}{2} + n = \frac{2n(2n-1)}{2} = T_{2n-1} 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 six 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. The On-Line Encyclopedia of Integer Sequences 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. The simplest example of this is the sequence of square triangular numbers. 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. Katayama proved that if three different integers , , and are all at least 3 and not equal to 6, then only finitely many numbers are simultaneously -gonal, -gonal, and -gonal. Katayama, Furuya, and Nishioka proved that if the integer is such that s=5 or 7\le s\le 12, then the only -gonal square triangular number is 1. For example, that paper gave the following proof for the case where s=5. Suppose that P(3,n)=P(4,p)=P(5,q) for some positive integers , , and . A calculation shows that the point (x,y) defined by (x,y)=(48p^{2}+3,24p(2n+1)(6q-1)) is on the curve Y^{2}=X^{3}-X^{2}-9X+9. That fact forces (x,y)=(51,360) (as an elliptic curve database confirms), so p=1 and the result follows. The number 1225 is hecatonicositetragonal (), hexacontagonal (), icosienneagonal (), hexagonal, square, and triangular. == See also ==