Polygonal number
In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers.
Contents
Definition and examples[edit]
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[edit]
Square numbers[edit]
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[edit]
Hexagonal numbers[edit]
Formula[edit]
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If s is the number of sides in a polygon, the formula for the n^{th} s-gonal number P(s,n) is
or
The n^{th} s-gonal number is also related to the triangular numbers T_{n} as follows:
Thus:
For a given s-gonal number P(s,n) = x, one can find n by
Every hexagonal number is also a triangular number[edit]
Applying the formula above:
to the case of 6 sides gives:
but since:
it follows that:
This shows that the hexagonal number, is the 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[edit]
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.^{[1]}
s | Name | Formula | n | Sum of Reciprocals^{[1]}^{[2]} | OEIS number | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||||
3 | Triangular | ½(n²+n) | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 | ^{[1]} | A000217 |
4 | Square | n² = ½(2n² - 0n) | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | ^{[1]} | A000290 |
5 | Pentagonal | ½(3n² - n) | 1 | 5 | 12 | 22 | 35 | 51 | 70 | 92 | 117 | 145 | ^{[1]} | A000326 |
6 | Hexagonal | ½(4n² - 2n) | 1 | 6 | 15 | 28 | 45 | 66 | 91 | 120 | 153 | 190 | ^{[1]} | A000384 |
7 | Heptagonal | ½(5n² - 3n) | 1 | 7 | 18 | 34 | 55 | 81 | 112 | 148 | 189 | 235 | ^{[1]} | A000566 |
8 | Octagonal | ½(6n² - 4n) | 1 | 8 | 21 | 40 | 65 | 96 | 133 | 176 | 225 | 280 | ^{[1]} | A000567 |
9 | Nonagonal | ½(7n² - 5n) | 1 | 9 | 24 | 46 | 75 | 111 | 154 | 204 | 261 | 325 | A001106 | |
10 | Decagonal | ½(8n² - 6n) | 1 | 10 | 27 | 52 | 85 | 126 | 175 | 232 | 297 | 370 | A001107 | |
11 | Hendecagonal | ½(9n² - 7n) | 1 | 11 | 30 | 58 | 95 | 141 | 196 | 260 | 333 | 415 | A051682 | |
12 | Dodecagonal | ½(10n² - 8n) | 1 | 12 | 33 | 64 | 105 | 156 | 217 | 288 | 369 | 460 | A051624 | |
13 | Tridecagonal | ½(11n² - 9n) | 1 | 13 | 36 | 70 | 115 | 171 | 238 | 316 | 405 | 505 | A051865 | |
14 | Tetradecagonal | ½(12n² - 10n) | 1 | 14 | 39 | 76 | 125 | 186 | 259 | 344 | 441 | 550 | A051866 | |
15 | Pentadecagonal | ½(13n² - 11n) | 1 | 15 | 42 | 82 | 135 | 201 | 280 | 372 | 477 | 595 | A051867 | |
16 | Hexadecagonal | ½(14n² - 12n) | 1 | 16 | 45 | 88 | 145 | 216 | 301 | 400 | 513 | 640 | A051868 | |
17 | Heptadecagonal | ½(15n² - 13n) | 1 | 17 | 48 | 94 | 155 | 231 | 322 | 428 | 549 | 685 | A051869 | |
18 | Octadecagonal | ½(16n² - 14n) | 1 | 18 | 51 | 100 | 165 | 246 | 343 | 456 | 585 | 730 | A051870 | |
19 | Enneadecagonal | ½(17n² - 15n) | 1 | 19 | 54 | 106 | 175 | 261 | 364 | 484 | 621 | 775 | A051871 | |
20 | Icosagonal | ½(18n² - 16n) | 1 | 20 | 57 | 112 | 185 | 276 | 385 | 512 | 657 | 820 | A051872 | |
21 | Icosihenagonal | ½(19n² - 17n) | 1 | 21 | 60 | 118 | 195 | 291 | 406 | 540 | 693 | 865 | A051873 | |
22 | Icosidigonal | ½(20n² - 18n) | 1 | 22 | 63 | 124 | 205 | 306 | 427 | 568 | 729 | 910 | A051874 | |
23 | Icositrigonal | ½(21n² - 19n) | 1 | 23 | 66 | 130 | 215 | 321 | 448 | 596 | 765 | 955 | A051875 | |
24 | Icositetragonal | ½(22n² - 20n) | 1 | 24 | 69 | 136 | 225 | 336 | 469 | 624 | 801 | 1000 | A051876 | |
... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... |
10000 | Myriagonal | ½(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
Combinations[edit]
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 | Odd triangular square numbers. | 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, 4835857349623374369, 571178264921393749929, 4342594514813297471521, 512917445842648529510881, … | 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 |
10 | 3 | 1, 10, 1540, 1777555, 13773376, 2051297326, 15894464365, 2367195337045, 18342198104230, ... | |
10 | 4 | 1 and no other. | |
11 | 4 | 1, 196, 29241, 1755625, 261468900, 38941102225, 2337990844401, 348201795147556, 51858411008887561, 3113535139359330841, 463705205422871375236, 69060571958250748760481, 4146338334574433921200225, ... | |
12 | 3 | 1, 105, 561, 41041, 3000025, 219294153, 1167483681,... | |
12 | 4 | 1, 64, 3025, 142129, 6677056, 313679521, 14736260449, 692290561600, 32522920134769, 1527884955772561, 71778070001175616, 3372041405099481409, 158414167969674450625, ... | |
12 | 5 | 1, 12, 324105, 2884960, 75296819577, ... | |
13 | 3 | 1, 36, 171, 4186, 237016, 5785101, 27169506, 663153571, 37543891231, ... | |
13 | 4 | 1, 36, 35721, 34999056, 896703025, 34291262041, 878568782400, 860801272542225, ... | |
14 | 3 | 1, 1275, 58311, 48005301, 2194498125, ... | |
14 | 4 | 1, 441, 14161, 4239481, 135978921, 40707501121, 1305669590281, 390873421529361, 12537039269904241, 3753166552817428201, ... | |
15 | 3 | 1, 15, 595, 221445, 863955, 321197185, 12232747905, 119413111951, 4547830928851, 44394820201725, ... | |
15 | 4 | 1, 3025, 5997601, 165148201, 327372787225, 648950099301001, 17869299978562201, ... | |
16 | 3 | 1, 45, 1485, 41041, 1334161, 36855405, 1198075725, ... | |
16 | 4 | 1, 16, 400, 4225, 101761, 1073296, 25847056, 272613121, 6565050625, ... | |
18 | 4 | 1, 100, 1936, 116281, 2235025, 134189056, 2579217796, 154854055225, 2976415102441, ... | |
21 | 3 | 1, 21, 406, 30381, 86736, 6474601, 119761026, 1379779246, ... | |
21 | 4 | 1, 2601, 3705625, 54745201, ... | |
22 | 3 | 1, 11935, 203203, 1237556125, 21068604901, ... | |
22 | 4 | 1, 729, 284089, 3900625, 15175959521, 590725976569, 8110813506601, 3156387347610225, 1228333148092290241, 16865317394711073289, 6563271907899976822281, 2554149271482890096235025, 35069100108493095964960369, ... | |
23 | 3 | 1, 66, 558096, 18644671, 294116131, 9825706836, ... | |
24 | 3 | 1, 136, 2016, 51681, 134421, 3444000, 50637016, 3374352325, ... | |
24 | 4 | 1, 225, 25921, 312481, 35820225, 4106118241, ... | |
25 | 3 | 1, 351, 4851, 810901, 11169901, 1866696651, ... | |
25 | 4 | 1, 25, 8100, 104976, 32478601, 420865225, 4786564225, 62025404401, ... | |
28 | 3 | 1, 28, 4465, 10585, 1606528, 20985481, 243796321, 3184618528, ... | |
28 | 4 | 1, 81, 3136, 30625, 1166400, 44422225, 433638976, 16515077121, ... | |
29 | 3 | 1, 1225, 2850, 1519896, 16799706, 8954498400, ... | |
29 | 4 | 1, 1225, 680625, 377719225, 209618381281, ... | |
30 | 3 | 1, 595, 3649051, 926844985, 9775585225, ... | |
30 | 4 | 1, 203401, 1819801, 164024190001, 1467492382801, 132269434866199801, 1183388792474889001, 106662336814809228952801, 954287089027867949018401, 86012721732003522411131649001, 769539017165067381031862931001, ... | |
32 | 4 | 1, 1089, 9025, 4190209, 34680321, 16098788161, 133241790529, 61851539930625, 511914924538369, 237633600314679361, 1966777006834629441, 912988230557458180609, 7556356748343721780225, ... | |
34 | 3 | 1, 325, 2926, 377146, 3378700, 435228256, ... | |
36 | 3 | 1, 36, 105, 1540, 12880, 185745, 523776, 7552441, 63095761, 909788496, 2565464265, ... | |
36 | 4 | 1, 36, 4096, 443556, 6630625, 48011041, 717704100, 77684838400, 8408666052900, 125698875825601, 910160415388225, ... | |
40 | 4 | 1, 576, 123201, 26255376, 5595189601, ... | |
41 | 3 | 1, 120, 57970, 2478651, 17841351, 762822270, ... | |
44 | 4 | 1, 256, 1521, 136161, 802816, 71757841, 423083761, 37816247296, 222964340481, ... | |
45 | 3 | 1, 45, 435, 2850, 44551, 41810940, 79714251, ... | |
49 | 3 | 1, 4950, 29646, 45634681, 273183625, ... | |
49 | 4 | 1, 49, 144, 12996, 613089, 54331641, 2562586884, 227094183936, 644826666121, ... | |
50 | 3 | 1, 20910, 122760, 787032975, 4620055875, ... | |
50 | 4 | 1, 5776, 30276, 55487601, 290736601, 532791965476, 2791652838976, 5115868397039401, 26805450269137401, 49122567815580389376, 257385930692604511876, 471674891049334501775401, 2471419679704938253922401, ... |
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 appear in print.^{[3]}
The number 1225 is hecatonicositetragonal (s=124), hexacontagonal (s=60), icosienneagonal (s=29), hexagonal, square, and triangular.
The only polygonal set that is contained in another polygonal set is the set of hexagonal numbers, which is contained in the set of triangular numbers.^{[citation needed]}
See also[edit]
Notes[edit]
References[edit]
- The Penguin Dictionary of Curious and Interesting Numbers, David Wells (Penguin Books, 1997) [ISBN 0-14-026149-4].
- Polygonal numbers at PlanetMath
- Weisstein, Eric W. "Polygonal Numbers". MathWorld.
- F. Tapson (1999). The Oxford Mathematics Study Dictionary (2nd ed.). Oxford University Press. pp. 88–89. ISBN 0-19-914-567-9.
External links[edit]
- Hazewinkel, Michiel, ed. (2001) [1994], "Polygonal number", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Polygonal Numbers: Every s-polygonal number between 1 and 1000 clickable for 2<=s<=337
- Polygonal Numbers on the Ulam Spiral grid on YouTube
- Polygonal Number Counting Function: http://www.mathisfunforum.com/viewtopic.php?id=17853