- Polygonal number
In
mathematics , a polygonal number is anumber that can be arranged as a regularpolygon . Ancientmathematician s discovered that numbers could be arranged in certain ways when they were represented by pebbles or seeds; such numbers, which can be made from figures, are generally calledfigurate number s.The number 10, for example, can be arranged as a
triangle (seetriangular 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
If "s" is the number of sides in a polygon, the formula for the "n"th "s"-gonal number is s-2)n^2-(s-4)n}over 2.
Name Formula "n"=1 2 3 4 5 6 7 8 9 10 11 12 13 Triangular ½(1"n"² + 1"n") 1 3 6 10 15 21 28 36 45 55 66 78 91 Square ½(2"n"² - 0"n") 1 4 9 16 25 36 49 64 81 100 121 144 169 Pentagonal ½(3"n"² - 1"n") 1 5 12 22 35 51 70 92 117 145 176 210 247 Hexagonal ½(4"n"² - 2"n") 1 6 15 28 45 66 91 120 153 190 231 276 325 Heptagonal ½(5"n"² - 3"n") 1 7 18 34 55 81 112 148 189 235 286 342 403 Octagonal ½(6"n"² - 4"n") 1 8 21 40 65 96 133 176 225 280 341 408 481 Nonagonal ½(7"n"² - 5"n") 1 9 24 46 75 111 154 204 261 325 396 474 559 Decagonal ½(8"n"² - 6"n") 1 10 27 52 85 126 175 232 297 370 451 540 637 Hendecagonal ½(9"n"² - 7"n") 1 11 30 58 95 141 196 260 333 415 506 606 715 Dodecagonal ½(10"n"² - 8"n") 1 12 33 64 105 156 217 288 369 460 561 672 793 Tridecagonal ½(11"n"² - 9"n") 1 13 36 70 115 171 238 316 405 505 616 738 871 Tetradecagonal ½(12"n"² - 10"n") 1 14 39 76 125 186 259 344 441 550 671 804 949 Pentadecagonal ½(13"n"² - 11"n") 1 15 42 82 135 201 280 372 477 595 726 870 1027 Hexadecagonal ½(14"n"² - 12"n") 1 16 45 88 145 216 301 400 513 640 781 936 1105 Heptadecagonal ½(15"n"² - 13"n") 1 17 48 94 155 231 322 428 549 685 836 1002 1183 Octadecagonal ½(16"n"² - 14"n") 1 18 51 100 165 246 343 456 585 730 891 1068 1261 Nonadecagonal ½(17"n"² - 15"n") 1 19 54 106 175 261 364 484 621 775 946 1134 1339 Icosagonal ½(18"n"² - 16"n") 1 20 57 112 185 276 385 512 657 820 1001 1200 1417 Icosihenagonal ½(19"n"² - 17"n") 1 21 60 118 195 291 406 540 693 865 1056 1266 1495 Icosidigonal ½(20"n"² - 18"n") 1 22 63 124 205 306 427 568 729 910 1111 1332 1573 Icositrigonal ½(21"n"² - 19"n") 1 23 66 130 215 321 448 596 765 955 1166 1398 1651 Icositetragonal ½(22"n"² - 20"n") 1 24 69 136 225 336 469 624 801 1000 1221 1464 1729 Icosipentagonal ½(23"n"² - 21"n") 1 25 72 142 235 351 490 652 837 1045 1276 1530 1807 Icosihexagonal ½(24"n"² - 22"n") 1 26 75 148 245 366 511 680 873 1090 1331 1596 1885 Icosiheptagonal ½(25"n"² - 23"n") 1 27 78 154 255 381 532 708 909 1135 1386 1662 1963 Icosioctagonal ½(26"n"² - 24"n") 1 28 81 160 265 396 553 736 945 1180 1441 1728 2041 Icosinonagonal ½(27"n"² - 25"n") 1 29 84 166 275 411 574 764 981 1225 1496 1794 2119 Triacontagonal ½(28"n"² - 26"n") 1 30 87 172 285 426 595 792 1017 1270 1551 1860 2197 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").For a given "s"-gonal number "x", one can find "n" by
:n = frac{sqrt{8(s-2)x+(s-4)^2}+s-4}{2(s-2)}.
References
*"The Penguin Dictionary of Curious and Interesting Numbers", David Wells (Penguin Books, 1997) [ISBN 0-14-026149-4] .
* [http://planetmath.org/encyclopedia/PolygonalNumber.html Polygonal numbers at PlanetMath]
* [http://mathworld.wolfram.com/PolygonalNumber.html Polygonal numbers at MathWorld]External links
* [http://www.virtuescience.com/polygonal-numbers.html Polygonal Numbers: Every polygonal number between 1 and 1000 clickable]
*youtube|id=YOiZ459lZ7A|title=Polygonal Numbers on the Ulam Spiral grid
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