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Figurate numbers

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Since at least the time of the ancient Greeks, man has studied sequences of numbers which correspond to the geometric arrangement of like items (e.g. dots, pebbles, atoms, ...) which begets the figurate numbers.[1] For the Greeks of Antiquity, these numbers related either to 2-dimensional figures such as polygons,[2] or to 3-dimensional figures such as pyramids.[3] Figurate numbers, which are usually associated with the 2 or 3-dimensional figures, may be generalized to higher dimensions where they are often called polytope numbers and also to lower dimensions (where they are called gnomonic numbers and centered gnomonic numbers for 1-dimensional figures).

This introductory page shows only the 2-dimensional regular convex figurate numbers (polygonal numbers and centered polygonal numbers) and their associated 3-dimensional pyramidal layerings (pyramidal numbers and centered pyramidal numbers, i.e. (centered polygons) pyramidal numbers), which are not globally centered (only the original polygons are). Although pyramids (with any polygonal base) are not regular polyhedrons (with the exception of triangular pyramids, which are tetrahedrons,[4] one of the 5 Platonic solids[5]), the Greeks had a particular fascination towards square pyramids.[6] It turns out that a square dipyramid[7] is a regular polyhedron, an octahedron,[8] one of the 5 Platonic solids!

All figurate numbers are fully classified on the page: Classifications of figurate numbers.

Two-dimensional figurate numbers

Polygonal numbers

     Triangular numbers           Square numbers           Pentagonal numbers           Hexagonal numbers     [9]
Triangular numbers Square numbers Pentagonal numbers Hexagonal numbers
The polygonal numbers
P  (2)V(n)
model
V
-sided (the number of sides of a polygon being equal to its number of vertices
V  
) regular convex polygons which grow by the addition of layers to
V  −  2
of the sides.

The general formula for regular convex polygonal numbers is:[10]

P  (2)V(n)  = 
(V − 2) n 2 − (V − 4) n
2
 = 
n
2
[(V − 2) n − (V − 4)],
where
V
is the number of vertices (thus sides) of the polygon. Prime numbers (shown in bold) occur only in
P  (2)V(2) = V
since
P  (2)V(n), n   ≥   3,
is obviously composite from the formula.

See Polygonal numbers (and Category:Polygonal numbers) for more information (e.g. more formulae, recurrence equations, generating functions, order of basis, differences, partial sums, partial sums of reciprocals, sum of reciprocals) about the regular polygonal numbers.

Polygonal numbers formulae and values
A-number
V
-gonal numbers
Formula
P  (2)V(n)
n = 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A000217
 (n)
Trigonal numbers
(Triangular numbers)
0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 210
A000290
 (n)
Tetragonal numbers
(Square numbers)
0 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400
A000326
 (n)
Pentagonal
numbers
0 1 5 12 22 35 51 70 92 117 145 176 210 247 287 330 376 425 477 532 590
A000384
 (n)
Hexagonal
numbers
0 1 6 15 28 45 66 91 120 153 190 231 276 325 378 435 496 561 630 703 780
A000566
 (n)
Heptagonal
numbers
0 1 7 18 34 55 81 112 148 189 235 286 342 403 469 540 616 697 783 874 970
A000567
 (n)
Octagonal
numbers
0 1 8 21 40 65 96 133 176 225 280 341 408 481 560 645 736 833 936 1045 1160
A001106
 (n)
Nonagonal
numbers
0 1 9 24 46 75 111 154 204 261 325 396 474 559 651 750 856 969 1089 1216 1350
A001107
 (n)
Decagonal
numbers
0 1 10 27 52 85 126 175 232 297 370 451 540 637 742 855 976 1105 1242 1387 1540
A051682
 (n)
Hendecagonal
numbers
0 1 11 30 58 95 141 196 260 333 415 506 606 715 833 960 1096 1241 1395 1558 1730
A051624
 (n)
Dodecagonal
numbers
0 1 12 33 64 105 156 217 288 369 460 561 672 793 924 1065 1216 1377 1548 1729 1920
A051865
 (n)
Tridecagonal
numbers
0 1 13 36 70 115 171 238 316 405 505 616 738 871 1015 1170 1336 1513 1701 1900 2110
A051866
 (n)
Tetradecagonal
numbers
0 1 14 39 76 125 186 259 344 441 550 671 804 949 1106 1275 1456 1649 1854 2071 2300
A051867
 (n)
Pentadecagonal
numbers
0 1 15 42 82 135 201 280 372 477 595 726 870 1027 1197 1380 1576 1785 2007 2242 2490
A051868
 (n)
Hexadecagonal
numbers
0 1 16 45 88 145 216 301 400 513 640 781 936 1105 1288 1485 1696 1921 2160 2413 2680
A051869
 (n)
Heptadecagonal
numbers
0 1 17 48 94 155 231 322 428 549 685 836 1002 1183 1379 1590 1816 2057 2313 2584 2870
A051870
 (n)
Octadecagonal
numbers
0 1 18 51 100 165 246 343 456 585 730 891 1068 1261 1470 1695 1936 2193 2466 2755 3060
A051871
 (n)
Nonadecagonal
numbers
0 1 19 54 106 175 261 364 484 621 775 946 1134 1339 1561 1800 2056 2329 2619 2926 3250
A051872
 (n)
Icosagonal
numbers
0 1 20 57 112 185 276 385 512 657 820 1001 1200 1417 1652 1905 2176 2465 2772 3097 3440

Centered polygonal numbers

     Centered triangular numbers           Centered square numbers           Centered pentagonal numbers           Centered hexagonal numbers     [9]
Centered triangular numbers Centered square numbers Centered pentagonal numbers Centered hexagonal numbers
(Hex numbers)
The centered polygonal numbers
cP  (2)V(n)
model
V
-sided regular convex polygons (the number of sides of a polygon being equal to its number of vertices
V  
) centered on a single dot (for
n   ≥   0
), in which each successive layer surrounds the previous layer.

The general formula for centered polygonal numbers is:[11]

cP  (2)V(n)  =  V P  (2)3(n) + 1  =  V Tn + 1  =  V( n + 1 2  ) + 1  =  V
n  (n + 1)
2
+ 1,
where
P  (2)3(n) = Tn
is the
n
th triangular number.

Prime numbers (shown in bold) occur often among the centered polygonal numbers due to the 1 addend in the formula. There are primes among all the centered polygonal numbers from the centered triangular numbers to the centered icosagonal numbers, with the notable exceptions of the centered octagonal numbers (which can never be prime since they are odd squares) and the centered nonagonal numbers (which can never be prime since they are a subsequence of the triangular numbers).

See Centered polygonal numbers (and Category:Centered polygonal numbers) for more information (e.g. more formulae, recurrence equations, generating functions, order of basis, differences, partial sums, partial sums of reciprocals, sum of reciprocals) about the centered polygonal numbers.

Centered polygonal numbers formulae and values
A-number Centered
V
-gonal
numbers
Formula
cP  (2)V(n)  = 
V  Tn + 1
n = 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 # of primes
for
n
in [0 .. 19]
A005448
 (n + 1)
Centered trigonal numbers
(Centered triangular numbers)
1 4 10 19 31 46 64 85 109 136 166 199 235 274 316 361 409 460 514 571 6
A001844
 (n)
Centered tetragonal numbers
(Centered square numbers)
1 5 13 25 41 61 85 113 145 181 221 265 313 365 421 481 545 613 685 761 10
A005891
 (n)
Centered
pentagonal numbers
1 6 16 31 51 76 106 141 181 226 276 331 391 456 526 601 681 766 856 951 4
A003215
 (n)
Centered hexagonal numbers
(Hex numbers)
1 7 19 37 61 91 127 169 217 271 331 397 469 547 631 721 817 919 1027 1141 11
A069099
 (n + 1)
Centered
heptagonal numbers
1 8 22 43 71 106 148 197 253 316 386 463 547 638 736 841 953 1072 1198 1331 6
A016754
 (n)
Centered
octagonal numbers

1 9 25 49 81 121 169 225 289 361 441 529 625 729 841 961 1089 1225 1369 1521 0
A060544
 (n + 1)
Centered
nonagonal numbers

1 10 28 55 91 136 190 253 325 406 496 595 703 820 946 1081 1225 1378 1540 1711 0
A062786
 (n + 1)
Centered
decagonal numbers
1 11 31 61 101 151 211 281 361 451 551 661 781 911 1051 1201 1361 1531 1711 1901 14
A069125
 (n + 1)
Centered
hendecagonal numbers
1 12 34 67 111 166 232 309 397 496 606 727 859 1002 1156 1321 1497 1684 1882 2091 5
A003154
 (n + 1)
Centered
dodecagonal numbers
1 13 37 73 121 181 253 337 433 541 661 793 937 1093 1261 1441 1633 1837 2053 2281 12
A069126
 (n + 1)
Centered
tridecagonal numbers
1 14 40 79 131 196 274 365 469 586 716 859 1015 1184 1366 1561 1769 1990 2224 2471 3
A069127
 (n + 1)
Centered
tetradecagonal numbers
1 15 43 85 141 211 295 393 505 631 771 925 1093 1275 1471 1681 1905 2143 2395 2661 6
A069128
 (n + 1)
Centered
pentadecagonal numbers
1 16 46 91 151 226 316 421 541 676 826 991 1171 1366 1576 1801 2041 2296 2566 2851 7
A069129
 (n + 1)
Centered
hexadecagonal numbers
1 17 49 97 161 241 337 449 577 721 881 1057 1249 1457 1681 1921 2177 2449 2737 3041 9
A069130
 (n + 1)
Centered
heptadecagonal numbers
1 18 52 103 171 256 358 477 613 766 936 1123 1327 1548 1786 2041 2313 2602 2908 3231 4
A069131
 (n + 1)
Centered
octadecagonal numbers
1 19 55 109 181 271 379 505 649 811 991 1189 1405 1639 1891 2161 2449 2755 3079 3421 9
A069132
 (n + 1)
Centered
nonadecagonal numbers
1 20 58 115 191 286 400 533 685 856 1046 1255 1483 1730 1996 2281 2585 2908 3250 3611 3
A069133
 (n + 1)
Centered
icosagonal numbers
1 21 61 121 201 301 421 561 721 901 1101 1321 1561 1821 2101 2401 2721 3061 3421 3801 4

Three-dimensional figurate numbers

Pyramidal numbers

The pyramidal numbers
Y  (3)V(n)
model
(V  −  1)
-gonal (the number of sides of the polygonal base of a pyramid being equal to its number of vertices
V
minus one for the apex vertex) pyramids in which each horizontal layer corresponds to a regular convex polygonal number, thus:
Y  (3)V(n)  ≡ 
n
i   = 0
  
P  (2)[V −1] (i).

The general formula for the pyramidal numbers is:[12]

Y  (3)V(n)  = 
([V − 1] − 2) n 3 + 3 n 2 − ([V − 1] − 5) n
6
 = 
n
6
{([V − 1] − 2) n 2 + 3 n − ([V − 1] − 5)}.
Prime numbers (shown in bold) may appear only for
Y  (3)V(2) = V + 1
since
Y  (3)V(n), n   ≥   3,
is obviously composite from the formula.

See Pyramidal numbers (and Category:Pyramidal numbers) for more information (e.g. more formulae, recurrence equations, generating functions, order of basis, differences, partial sums, partial sums of reciprocals, sum of reciprocals) about the pyramidal numbers.

Pyramidal numbers formulae and values
A-number
(V  −  1)
-gonal
pyramidal numbers
Formula
Y  (3)V(n)
n = 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A000292
 (n)
Trigonal pyramidal numbers
(Tetrahedral numbers)

0 1 4 10 20 35 56 84 120 165 220 286 364 455 560 680 816 969 1140 1330 1540
A000330
 (n)
Tetragonal pyramidal numbers
(Square pyramidal numbers)


0 1 5 14 30 55 91 140 204 285 385 506 650 819 1015 1240 1496 1785 2109 2470 2870
A002411
 (n)
Pentagonal
pyramidal numbers
0 1 6 18 40 75 126 196 288 405 550 726 936 1183 1470 1800 2176 2601 3078 3610 4200
A002412
 (n)
Hexagonal
pyramidal numbers
0 1 7 22 50 95 161 252 372 525 715 946 1222 1547 1925 2360 2856 3417 4047 4750 5530
A002413
 (n)
Heptagonal
pyramidal numbers
0 1 8 26 60 115 196 308 456 645 880 1166 1508 1911 2380 2920 3536 4233 5016 5890 6860
A002414
 (n)
Octagonal
pyramidal numbers
0 1 9 30 70 135 231 364 540 765 1045 1386 1794 2275 2835 3480 4216 5049 5985 7030 8190
A007584
 (n)
Nonagonal
pyramidal numbers
0 1 10 34 80 155 266 420 624 885 1210 1606 2080 2639 3290 4040 4896 5865 6954 8170 9520
A007585
 (n)
Decagonal
pyramidal numbers
0 1 11 38 90 175 301 476 708 1005 1375 1826 2366 3003 3745 4600 5576 6681 7923 9310 10850
A007586
 (n)
Hendecagonal
pyramidal numbers
0 1 12 42 100 195 336 532 792 1125 1540 2046 2652 3367 4200 5160 6256 7497 8892 10450 12180
A007587
 (n)
Dodecagonal
pyramidal numbers
0 1 13 46 110 215 371 588 876 1245 1705 2266 2938 3731 4655 5720 6936 8313 9861 11590 13510
A050441
 (n)
Tridecagonal
pyramidal numbers
0 1 14 50 120 235 406 644 960 1365 1870 2486 3224 4095 5110 6280 7616 9129 10830 12730 14840
A172073
 (n)
Tetradecagonal
pyramidal numbers
0 1 15 54 130 255 441 700 1044 1485 2035 2706 3510 4459 5565 6840 8296 9945 11799 13870 16170
A??????
 (n)
Pentadecagonal
pyramidal numbers
0 1 16 58 140 275 476 756 1128 1605 2200 2926 3796 4823 6020 7400 8976 10761 12768 15010 17500
A172076
 (n)
Hexadecagonal
pyramidal numbers
0 1 17 62 150 295 511 812 1212 1725 2365 3146 4082 5187 6475 7960 9656 11577 13737 16150 18830
A??????
 (n)
Heptadecagonal
pyramidal numbers
0 1 18 66 160 315 546 868 1296 1845 2530 3366 4368 5551 6930 8520 10336 12393 14706 17290 20160
A172078
 (n)
Octadecagonal
pyramidal numbers
0 1 19 70 170 335 581 924 1380 1965 2695 3586 4654 5915 7385 9080 11016 13209 15675 18430 21490
A??????
 (n)
Nonadecagonal
pyramidal numbers
0 1 20 74 180 355 616 980 1464 2085 2860 3806 4940 6279 7840 9640 11696 14025 16644 19570 22820
A172082
 (n)
Icosagonal
pyramidal numbers
0 1 21 78 190 375 651 1036 1548 2205 3025 4026 5226 6643 8295 10200 12376 14841 17613 20710 24150

Centered pyramidal numbers

The centered pyramidal numbers, i.e. (centered polygons) pyramidal numbers are a family of sequences of 3-dimensional nonregular polytope numbers (among the 3-dimensional figurate numbers) formed by adding the first
[N0  −  1]
positive centered polygonal numbers with constant number of sides
[N0  −  1]
, where
N0
is the number of vertices (including the apex vertex) of the polygonal base pyramid. The term centered pyramid numbers, i.e. (centered squares) pyramidal numbers, is often used to refer to the centered square pyramidal numbers, i.e. (centered squares) pyramidal numbers, having a polygonal base with four sides. The centered pyramidal numbers, i.e. (centered polygons) pyramidal numbers, are a generalization of the centered pyramid numbers, i.e. (centered squares) pyramidal numbers, where the base is a regular convex polygon with any number of sides
[N0  −  1]   ≥   3
. Centered pyramidal numbers, i.e. (centered polygons) pyramidal numbers, may also be generalized to higher dimensions as centered hyperpyramidal numbers, i.e. (centered polygons) hyperpyramidal numbers. While the (centered polygons) pyramidal numbers are pyramidal stacks of centered polygons, the generated figures are not globally centered. Thus the (centered polygons) pyramidal numbers do not belong to the category of globally centered figurate numbers (which start with the globally central dot, giving value 1, for
n = 0
), they belong to the category of globally noncentered figurate numbers (which equals 0 for
n = 0
) and start with the initial dot, giving value 1, for
n = 1
). It would be less confusing if the centered pyramidal numbers were called (centered polygons) pyramidal numbers. Note that although the triangular pyramidal numbers are tetrahedral numbers, the (centered triangles) pyramidal numbers are NOT the (globally centered) centered tetrahedral numbers. The centered pyramidal numbers
‘c’Y  (3)V(n)
model
(V  −  1)
-gonal (the number of sides of the polygonal base of a pyramid being equal to the number of vertices
V
minus one for the apex vertex) pyramids in which each horizontal layer corresponds to a centered polygonal number, thus
‘c’Y  (3)V(n)  ≡ 
n
i   = 0
  
cP  (2)[V −1] (i).
[13]

The general formula for centered pyramidal numbers is

‘c’Y  (3)V(n)  = 
[V − 1] n 3 − ([V − 1] − 6) n
6
 = 
[V − 1] (n − 1) n (n + 1)
6
+ n  =  [V − 1] ( n + 1 3  ) + n.
Observe that the
n
th centered hexagonal pyramidal number is equal to the
n
th cube. Also the
n
th centered square pyramidal number is equal to the
n
th octahedral number, which is interesting since the
n
th octahedral number is the
n
th square dipyramidal number which is the sum of the
n
th square pyramidal number and the
(n  −  1)
th square pyramidal number. While the centered pyramidal numbers are pyramidal stacks of centered polygons, the generated figures are not globally centered. Thus the centered pyramidal numbers do not belong to the category of globally centered figurate numbers (which start with the globally central dot, giving value 1, for
n = 0
), they belong to the category of globally noncentered figurate numbers (which equals 0 for
n = 0
and start with the initial dot, giving value 1, for
n = 1
). It would be less confusing if the centered pyramidal numbers were called centered polygon pyramidal numbers. Note that although the triangular pyramidal numbers are tetrahedral numbers, the (centered polygons) centered triangular pyramidal numbers are NOT the (globally centered) centered tetrahedral numbers.

See Centered pyramidal numbers (and Category:Centered pyramidal numbers) for more information (e.g. more formulae, recurrence equations, generating functions, order of basis, differences, partial sums, partial sums of reciprocals, sum of reciprocals) about the centered pyramidal numbers.

Centered pyramidal numbers formulae and values
A-number Centered
(V  −  1)
-gonal
pyramidal numbers
Formula
‘c’Y  (3)V(n)
n = 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A006003
 (n)
Centered trigonal pyramidal numbers
(Centered triangular pyramidal numbers)
0 1 5 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010
A005900
 (n)
Centered tetragonal pyramidal numbers
(Centered square pyramidal numbers)


Octahedral numbers
0 1 6 19 44 85 146 231 344 489 670 891 1156 1469 1834 2255 2736 3281 3894 4579 5340
A004068
 (n)
Centered pentagonal pyramidal numbers 0 1 7 23 54 105 181 287 428 609 835 1111 1442 1833 2289 2815 3416 4097 4863 5719 6670
A000578
 (n)
Centered hexagonal pyramidal numbers
(Hex pyramidal numbers)


Cubes
0 1 8 27 64 125 216 343 512 729 1000 1331 1728 2197 2744 3375 4096 4913 5832 6859 8000
A004126
 (n)
Centered heptagonal pyramidal numbers 0 1 9 31 74 145 251 399 596 849 1165 1551 2014 2561 3199 3935 4776 5729 6801 7999 9330
A000447
 (n)
Centered octagonal pyramidal numbers 0 1 10 35 84 165 286 455 680 969 1330 1771 2300 2925 3654 4495 5456 6545 7770 9139 10660
A004188
 (n)
Centered nonagonal pyramidal numbers 0 1 11 39 94 185 321 511 764 1089 1495 1991 2586 3289 4109 5055 6136 7361 8739 10279 11990
A004466
 (n)
Centered decagonal pyramidal numbers 0 1 12 43 104 205 356 567 848 1209 1660 2211 2872 3653 4564 5615 6816 8177 9708 11419 13320
A004467
 (n)
Centered hendecagonal pyramidal numbers 0 1 13 47 114 225 391 623 932 1329 1825 2431 3158 4017 5019 6175 7496 8993 10677 12559 14650
A007588
 (n)
Centered dodecagonal pyramidal numbers

Stellated octahedron number[14]
(Stella octangula number) [15]
0 1 14 51 124 245 426 679 1016 1449 1990 2651 3444 4381 5474 6735 8176 9809 11646 13699 15980
A062025
 (n)
Centered tridecagonal pyramidal numbers 0 1 15 55 134 265 461 735 1100 1569 2155 2871 3730 4745 5929 7295 8856 10625 12615 14839 17310
A063521
 (n)
Centered tetradecagonal pyramidal numbers 0 1 16 59 144 285 496 791 1184 1689 2320 3091 4016 5109 6384 7855 9536 11441 13584 15979 18640
A063522
 (n)
Centered pentadecagonal pyramidal numbers 0 1 17 63 154 305 531 847 1268 1809 2485 3311 4302 5473 6839 8415 10216 12257 14553 17119 19970
A063523
 (n)
Centered hexadecagonal pyramidal numbers 0 1 18 67 164 325 566 903 1352 1929 2650 3531 4588 5837 7294 8975 10896 13073 15522 18259 21300
A??????
 (n)
Centered heptadecagonal pyramidal numbers 0 1 19 71 174 345 601 959 1436 2049 2815 3751 4874 6201 7749 9535 11576 13889 16491 19399 22630
A??????
 (n)
Centered octadecagonal pyramidal numbers 0 1 20 75 184 365 636 1015 1520 2169 2980 3971 5160 6565 8204 10095 12256 14705 17460 20539 23960
A??????
 (n)
Centered nonadecagonal pyramidal numbers 0 1 21 79 194 385 671 1071 1604 2289 3145 4191 5446 6929 8659 10655 12936 15521 18429 21679 25290
A??????
 (n)
Centered icosagonal pyramidal numbers 0 1 22 83 204 405 706 1127 1688 2409 3310 4411 5732 7293 9114 11215 13616 16337 19398 22819 26620

See also

Notes

  1. Weisstein, Eric W., Figurate Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/FigurateNumber.html]
  2. Weisstein, Eric W., Polygon, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Polygon.html]
  3. Weisstein, Eric W., Pyramid, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Pyramid.html]
  4. Weisstein, Eric W., Tetrahedron, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Tetrahedron.html]
  5. Weisstein, Eric W., Platonic Solid, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/PlatonicSolid.html]
  6. Weisstein, Eric W., Square Pyramid, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/SquarePyramid.html]
  7. Weisstein, Eric W., Dipyramid, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Dipyramid.html]
  8. Weisstein, Eric W., Octahedron, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Octahedron.html]
  9. 9.0 9.1 Author of the plots: Stefan Friedrich Birkner, License: Creative Commons Attribution-ShareAlike 3.0 Unported.
  10. Where
    P  (d)N0(n)
    is the
    d
    -dimensional regular convex polytope number, where
    N0
    is the number of 0-dimensional elements (i.e. vertices
    V  
    ) of the regular polytope.
  11. Where
    cP  (d)N0(n)
    is the
    d
    -dimensional centered regular convex polytope number, where
    N0
    is the number of 0-dimensional elements (i.e. vertices
    V  
    ) of the regular polytope.
  12. Where
    Y  (d)N0(n) = Y  (d)[(k  + 2) + (d  − 2)](n) = Y  (d)k +d (n), k   ≥   1, n   ≥   0,
    is the
    d
    -dimensional,
    d   ≥   0
    ,
    (k + 2)
    -gonal base (hyper)pyramidal number where, for
    d   ≥   2
    ,
    N0 = (k  +  2) + (d  −  2) = k + d
    is the number of vertices (including the
    d  −  2
    apex vertices) of the polygonal base (hyper)pyramid.
  13. Where
    ‘c’Y  (d)[(k  + 2) + (d  − 2)] (n) =  ‘c’Y  (d)k +d  (n), k   ≥   1, n   ≥   0,
    is the
    d
    -dimensional,
    d   ≥   0
    ,
    (k + 2)
    -gonal base (centered polygons) (hyper)pyramidal number where, for
    d   ≥   2, N0 = [(k  +  2) + (d  −  2)] = k + d
    is the number of vertices (including the
    d  −  2
    apex vertices) of the (centered polygonal base) (hyper)pyramid (the quoted
    ‘c’
    emphasizes that only the polygons are centered, not the whole figure).
  14. Weisstein, Eric W., Stellated Octahedron, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/StellatedOctahedron.html]
  15. Weisstein, Eric W., Stella Octangula Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/StellaOctangulaNumber.html]

External links