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.
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.
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.
If s is the number of sides in a polygon, the formula for the nth s-gonal number P(s,n) is
or
The nth s-gonal number is also related to the triangular numbers Tn as follows: [1]
Thus:
For a given s-gonal number P(s,n) = x, one can find n by
and one can find s by
Applying the formula above:
to the case of 6 sides gives:
but since:
it follows that:
This shows that the nth hexagonal number P(6,n) is also the (2n − 1)th triangular number T2n−1. We can find every hexagonal number by simply taking the odd-numbered triangular numbers: [1]
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]
s | Name | Formula | n | Sum of reciprocals [2] [3] | OEIS number | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||||
2 | Natural (line segment) | 1/2(0n2 + 2n) = n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ∞ (diverges) | A000027 |
3 | Triangular | 1/2(n2 + n) | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 | 2 [2] | A000217 |
4 | Square | 1/2(2n2 − 0n) = n2 | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | π2/6 [2] | A000290 |
5 | Pentagonal | 1/2(3n2 − n) | 1 | 5 | 12 | 22 | 35 | 51 | 70 | 92 | 117 | 145 | 3 ln 3 − π√3/3 [2] | A000326 |
6 | Hexagonal | 1/2(4n2 − 2n) = 2n2 - n | 1 | 6 | 15 | 28 | 45 | 66 | 91 | 120 | 153 | 190 | 2 ln 2 [2] | A000384 |
7 | Heptagonal | 1/2(5n2 − 3n) | 1 | 7 | 18 | 34 | 55 | 81 | 112 | 148 | 189 | 235 | [2] | A000566 |
8 | Octagonal | 1/2(6n2 − 4n) = 3n2 - 2n | 1 | 8 | 21 | 40 | 65 | 96 | 133 | 176 | 225 | 280 | 3/4 ln 3 + π√3/12 [2] | A000567 |
9 | Nonagonal | 1/2(7n2 − 5n) | 1 | 9 | 24 | 46 | 75 | 111 | 154 | 204 | 261 | 325 | A001106 | |
10 | Decagonal | 1/2(8n2 − 6n) = 4n2 - 3n | 1 | 10 | 27 | 52 | 85 | 126 | 175 | 232 | 297 | 370 | ln 2 + π/6 | A001107 |
11 | Hendecagonal | 1/2(9n2 − 7n) | 1 | 11 | 30 | 58 | 95 | 141 | 196 | 260 | 333 | 415 | A051682 | |
12 | Dodecagonal | 1/2(10n2 − 8n) | 1 | 12 | 33 | 64 | 105 | 156 | 217 | 288 | 369 | 460 | A051624 | |
13 | Tridecagonal | 1/2(11n2 − 9n) | 1 | 13 | 36 | 70 | 115 | 171 | 238 | 316 | 405 | 505 | A051865 | |
14 | Tetradecagonal | 1/2(12n2 − 10n) | 1 | 14 | 39 | 76 | 125 | 186 | 259 | 344 | 441 | 550 | 2/5 ln 2 + 3/10 ln 3 + π√3/10 | A051866 |
15 | Pentadecagonal | 1/2(13n2 − 11n) | 1 | 15 | 42 | 82 | 135 | 201 | 280 | 372 | 477 | 595 | A051867 | |
16 | Hexadecagonal | 1/2(14n2 − 12n) | 1 | 16 | 45 | 88 | 145 | 216 | 301 | 400 | 513 | 640 | A051868 | |
17 | Heptadecagonal | 1/2(15n2 − 13n) | 1 | 17 | 48 | 94 | 155 | 231 | 322 | 428 | 549 | 685 | A051869 | |
18 | Octadecagonal | 1/2(16n2 − 14n) | 1 | 18 | 51 | 100 | 165 | 246 | 343 | 456 | 585 | 730 | 4/7 ln 2 − √2/14 ln (3 − 2√2)+ π(1 + √2)/14 | A051870 |
19 | Enneadecagonal | 1/2(17n2 − 15n) | 1 | 19 | 54 | 106 | 175 | 261 | 364 | 484 | 621 | 775 | A051871 | |
20 | Icosagonal | 1/2(18n2 − 16n) | 1 | 20 | 57 | 112 | 185 | 276 | 385 | 512 | 657 | 820 | A051872 | |
21 | Icosihenagonal | 1/2(19n2 − 17n) | 1 | 21 | 60 | 118 | 195 | 291 | 406 | 540 | 693 | 865 | A051873 | |
22 | Icosidigonal | 1/2(20n2 − 18n) | 1 | 22 | 63 | 124 | 205 | 306 | 427 | 568 | 729 | 910 | A051874 | |
23 | Icositrigonal | 1/2(21n2 − 19n) | 1 | 23 | 66 | 130 | 215 | 321 | 448 | 596 | 765 | 955 | A051875 | |
24 | Icositetragonal | 1/2(22n2 − 20n) | 1 | 24 | 69 | 136 | 225 | 336 | 469 | 624 | 801 | 1000 | A051876 | |
... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... |
10000 | Myriagonal | 1/2(9998n2 − 9996n) | 1 | 10000 | 29997 | 59992 | 99985 | 149976 | 209965 | 279952 | 359937 | 449920 | A167149 |
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 A086270):
with
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 s-gonal t-gonal numbers for small values of s and t.
s | t | Sequence | OEIS number |
---|---|---|---|
4 | 3 | 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025, 63955431761796, 2172602007770041, 73804512832419600, 2507180834294496361, 85170343853180456676, 2893284510173841030625, 98286503002057414584576, 3338847817559778254844961, ... | A001110 |
5 | 3 | 1, 210, 40755, 7906276, 1533776805, 297544793910, 57722156241751, 11197800766105800, 2172315626468283465, … | A014979 |
5 | 4 | 1, 9801, 94109401, 903638458801, 8676736387298001, 83314021887196947001, 799981229484128697805801, ... | A036353 |
6 | 3 | All hexagonal numbers are also triangular. | A000384 |
6 | 4 | 1, 1225, 1413721, 1631432881, 1882672131025, 2172602007770041, 2507180834294496361, 2893284510173841030625, 3338847817559778254844961, 3853027488179473932250054441, ... | A046177 |
6 | 5 | 1, 40755, 1533776805, … | A046180 |
7 | 3 | 1, 55, 121771, 5720653, 12625478965, 593128762435, 1309034909945503, 61496776341083161, 135723357520344181225, 6376108764003055554511, 14072069153115290487843091, … | A046194 |
7 | 4 | 1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609, 350709705290025, 25635978392186449, 9976444135331412025, … | A036354 |
7 | 5 | 1, 4347, 16701685, 64167869935, … | A048900 |
7 | 6 | 1, 121771, 12625478965, … | A048903 |
8 | 3 | 1, 21, 11781, 203841, … | A046183 |
8 | 4 | 1, 225, 43681, 8473921, 1643897025, 318907548961, 61866420601441, 12001766689130625, 2328280871270739841, 451674487259834398561, 87622522247536602581025, 16998317641534841066320321, … | A036428 |
8 | 5 | 1, 176, 1575425, 234631320, … | A046189 |
8 | 6 | 1, 11781, 113123361, … | A046192 |
8 | 7 | 1, 297045, 69010153345, … | A048906 |
9 | 3 | 1, 325, 82621, 20985481, … | A048909 |
9 | 4 | 1, 9, 1089, 8281, 978121, 7436529, 878351769, 6677994961, 788758910641, 5996832038649, 708304623404049, 5385148492712041, 636056763057925561, ... | A036411 |
9 | 5 | 1, 651, 180868051, … | A048915 |
9 | 6 | 1, 325, 5330229625, … | A048918 |
9 | 7 | 1, 26884, 542041975, … | A048921 |
9 | 8 | 1, 631125, 286703855361, … | A048924 |
In some cases, such as s = 10 and t = 4, 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. [4]
The number 1225 is hecatonicositetragonal (s = 124), hexacontagonal (s = 60), icosienneagonal (s = 29), hexagonal, square, and triangular.
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 nth triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, starting with the 0th triangular number, is
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 32 and can be written as 3 × 3.
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes and different dimensions. The term can mean
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular and equilateral. Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.
Not to be confused with hexadecimal numbers.
In mathematics and combinatorics, a centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. The following figures illustrate this arrangement for the first four centered hexagonal 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 pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are 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.
A pyramidal number is the number of points in a pyramid with a polygonal base and triangular sides. The term often refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to a pyramid with any number of sides. The numbers of points in the base and in layers parallel to the base are given by polygonal numbers of the given number of sides, while the numbers of points in each triangular side is given by a triangular number. It is possible to extend the pyramidal numbers to higher dimensions.
A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The nth tetrahedral number, Ten, is the sum of the first n triangular numbers, that is,
An octagonal number is a figurate number that gives the number of points in a certain octagonal arrangement. The octagonal number for n is given by the formula 3n2 − 2n, with n > 0. The first few octagonal numbers are
In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broader topic of figurate numbers representing the numbers of points forming regular patterns within different shapes.
A pronic number is a number that is the product of two consecutive integers, that is, a number of the form . The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers; however, the term "rectangular number" has also been applied to the composite numbers.
In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a given city block distance of the center dot on a regular square lattice. While centered square numbers, like figurate numbers in general, have few if any direct practical applications, they are sometimes studied in recreational mathematics for their elegant geometric and arithmetic properties.
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 in the previous layer; so starting from the second polygonal layer, each layer of a centered k-gonal number contains k more dots than the previous layer.
A nonagonal number, or an enneagonal number, is a figurate number that extends the concept of triangular and square numbers to the nonagon. However, unlike the triangular and square numbers, the patterns involved in the construction of nonagonal numbers are not rotationally symmetrical. Specifically, the nth nonagonal number counts the dots in a pattern of n nested nonagons, all sharing a common corner, where the ith nonagon in the pattern has sides made of i dots spaced one unit apart from each other. The nonagonal number for n is given by the formula:
In number theory, a pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1, either from left to right or from right to left. It is named because it represents the number of 3-dimensional unit spheres which can be packed into a pentatope of increasing side lengths.
A centered decagonal number is a centered figurate number that represents a decagon with a dot in the center and all other dots surrounding the center dot in successive decagonal layers. The centered decagonal number for n is given by the formula
In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an (n+m)-polytope, where n and m are dimensions of 2 (polygon) or higher.
5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number.
In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be arranged into a square pyramid. Equivalently, which squares can be represented as the sum of consecutive squares, starting from 1.