Polygonal number explained

In mathematics, a polygonal number is a number that counts dots arranged in the shape of a regular polygon. These are one type of 2-dimensional figurate numbers.

Definition and examples

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.

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.

Hexagonal numbers

Formula

If is the number of sides in a polygon, the formula for the th -gonal number is

P(s,n)=

(s-2)n2-(s-4)n
2

or

P(s,n)=(s-2)

n(n-1)
2

+n

The th -gonal number is also related to the triangular numbers as follows:[1]

P(s,n)=(s-2)Tn-1+n=(s-3)Tn-1+Tn.

Thus:

\begin{align} P(s,n+1)-P(s,n)&=(s-2)n+1,\\ P(s+1,n)-P(s,n)&=Tn-1=

n(n-1)
2

,\\ P(s+k,n)-P(s,n)&=kTn-1=k

n(n-1)
2

. \end{align}

For a given -gonal number, one can find by

n=

\sqrt{8(s-2)x+{(s-4)
2}+(s-4)}{2(s-2)}

and one can find by

s=2+

2
n
x-n
n-1
.

Every hexagonal number is also a triangular number

Applying the formula above:

P(s,n)=(s-2)Tn-1+n

to the case of 6 sides gives:

P(6,n)=4Tn-1+n

but since:

Tn-1=

n(n-1)
2

it follows that:

P(6,n)=

4n(n-1)
2

+n=

2n(2n-1)
2

=T2n-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 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.[2]

NameFormulaSum of reciprocals[3] OEIS number
1 2 3 4 5 6 7 8 9 10
2Natural (line segment)12345678910∞ (diverges)
3Triangular13 6101521283645552
4Square149162536496481100
5Pentagonal15122235517092117145
6Hexagonal161528456691120153190
7Heptagonal1718345581112148189235

\begin{matrix} \tfrac{2}{3}ln5\ +\tfrac{{1}+\sqrt{5}}{3}ln\tfrac\sqrt{10-2\sqrt{5}}{2}\ +\tfrac{{1}-\sqrt{5}}{3}ln\tfrac\sqrt{10+2\sqrt{5}}{2}\\ +\tfrac{\pi\sqrt{25-10\sqrt{5}}}{15} \end{matrix}

8Octagonal1821406596133176225280
9Nonagonal19244675111154204261325
10Decagonal110275285126175232297370
11Hendecagonal111305895141196260333415
12Dodecagonal1123364105156217288369460
13Tridecagonal1133670115171238316405505
14Tetradecagonal1143976125186259344441550
15Pentadecagonal1154282135201280372477595
16Hexadecagonal1164588145216301400513640
17Heptadecagonal1174894155231322428549685
18Octadecagonal11851100165246343456585730
19Enneadecagonal11954106175261364484621775
20Icosagonal12057112185276385512657820
21Icosihenagonal12160118195291406540693865
22Icosidigonal12263124205306427568729910
23Icositrigonal12366130215321448596765955
24Icositetragonal124691362253364696248011000
.............................................
10000Myriagonal110000299975999299985149976209965279952359937449920

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):

2P(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 .

SequenceOEIS number
431, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025, 63955431761796, 2172602007770041, 73804512832419600, 2507180834294496361, 85170343853180456676, 2893284510173841030625, 98286503002057414584576, 3338847817559778254844961, ...
531, 210, 40755, 7906276, 1533776805, 297544793910, 57722156241751, 11197800766105800, 2172315626468283465, …
541, 9801, 94109401, 903638458801, 8676736387298001, 83314021887196947001, 799981229484128697805801, ...
63All hexagonal numbers are also triangular.
641, 1225, 1413721, 1631432881, 1882672131025, 2172602007770041, 2507180834294496361, 2893284510173841030625, 3338847817559778254844961, 3853027488179473932250054441, ...
651, 40755, 1533776805, …
731, 55, 121771, 5720653, 12625478965, 593128762435, 1309034909945503, 61496776341083161, 135723357520344181225, 6376108764003055554511, 14072069153115290487843091, …
741, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609, 350709705290025, 25635978392186449, 9976444135331412025, …
751, 4347, 16701685, 64167869935, …
761, 121771, 12625478965, …
831, 21, 11781, 203841, …
841, 225, 43681, 8473921, 1643897025, 318907548961, 61866420601441, 12001766689130625, 2328280871270739841, 451674487259834398561, 87622522247536602581025, 16998317641534841066320321, …
851, 176, 1575425, 234631320, …
861, 11781, 113123361, …
871, 297045, 69010153345, …
931, 325, 82621, 20985481, …
941, 9, 1089, 8281, 978121, 7436529, 878351769, 6677994961, 788758910641, 5996832038649, 708304623404049, 5385148492712041, 636056763057925561, ...
951, 651, 180868051, …
961, 325, 5330229625, …
971, 26884, 542041975, …
981, 631125, 286703855361, …

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.

See also

References

External links

Notes and References

  1. Book: Conway, John H. . The Book of Numbers . Guy . Richard . 2012-12-06 . Springer Science & Business Media . 978-1-4612-4072-3 . 38-41 . en . John Horton Conway . Richard K. Guy.
  2. Web site: Sums of Reciprocals of Polygonal Numbers and a Theorem of Gauss . 2010-06-13 . https://web.archive.org/web/20110615085610/http://www.siam.org/journals/problems/downloadfiles/07-003s.pdf . 2011-06-15 . dead .
  3. Web site: Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers . 2010-05-13 . 2013-05-29 . https://web.archive.org/web/20130529032918/http://www.math.psu.edu/sellersj/downey_ong_sellers_cmj_preprint.pdf . dead .