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Polygonal numbers
[1] | |||
Triangular numbers | Square numbers | Pentagonal numbers | Hexagonal numbers |
n |
N0 |
V |
n |
n |
n ≥ 1 |
n = 1 |
N1 |
E |
N0 |
V |
All figurate numbers are accessible via this structured menu: Classifications of figurate numbers.
Contents
- 1 Formulae
- 2 Sequences
- 3 Polygonal roots
- 4 Schläfli–Poincaré (convex) polytope formula
- 5 Recurrence equation
- 6 Generating function
- 7 Order of basis
- 8 Differences
- 9 Partial sums
- 10 Partial sums of reciprocals
- 11 Sum of reciprocals
- 12 Table of formulae and values
- 13 Table of related formulae and values
- 14 Table of sequences
- 15 Mysterious large overlap of terms of A176949 with terms in A140164
- 16 See also
- 17 Notes
- 18 External links
Formulae
Then |
N0 |
-
P (2) N0 (n) := n∑ i = 0
=(n − 1) n 2
[(N0 − 2) n − (N0 − 4)],n 2
P (1) N0 (n) |
n |
N0 |
N0 |
V |
Tn |
n |
Nontrivial polygonal numbers
A number which is non-trivially polygonal (a nontrivial polygonal number?) is a numbern |
k (mod tk − 1) |
3 ≤ tk − 1 < n |
n = j ⋅ tk − 1 + k |
k ≥ 3 |
j ≥ 1 |
m |
k |
m = j + 2 |
n =
|
For example, for
n = 45 |
-
, thus of order45 mod t3 − 1 = 45 mod 3 = 3
;k = 3 -
;45 mod t4 − 1 = 45 mod 6 = 3 -
, thus of order45 mod t5 − 1 = 45 mod 10 = 5
;k = 5 -
;45 mod t6 − 1 = 45 mod 15 = 0 -
;45 mod t7 − 1 = 45 mod 21 = 3 -
;45 mod t8 − 1 = 45 mod 28 = 17 -
, thus of order45 mod t9 − 1 = 45 mod 36 = 9
;k = 9
m =
|
m = 16, 6, 3, |
k = 3, 5, 9, |
m |
n |
3 ≤ m < n |
( : units; : primes; : missed composites)
|
3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
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32 |
|
3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | |
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61 | |||||||||||||||||||||
62 | |||||||||||||||||||||
63 | |||||||||||||||||||||
64 |
Composite numbers n which are not m-gonal numbers for 3 ≤ m < n
In the following, forn ≥ 3 |
n |
n |
k = 2 |
n |
m |
m < n |
k > 2 |
For [composite] positive integers
n |
n = P (2) m (k) |
k ≥ 3 |
n =
|
Since we are looking for solutions of
(m − 2) k 2 − (m − 4) k − 2 n = 0 |
m ≥ 3 |
k ≥ 3 |
k |
k =
|
m = 3 |
-
3 ≤ k ≤ .−1 + √ 1 + 8 n2
2 n = m k (k − 1) − 2 k (k − 2) |
m ≥ 3 |
k ≥ 3 |
m |
m =
|
k = 3 |
-
3 ≤ m ≤ .n + 3 3
n |
(n) = n |
n |
m |
3 ≤ m < n |
- {4, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 77, 80, 86, 98, 104, 110, 116, 119, 122, 128, 134, 140, 143, 146, 152, 158, 161, 164, 170, 182, 187, 188, 194, 200, 203, 206, 209, 212, 218, 221, 224, 230, 236, 242, 248, 254, 266, 272, 278, 284, 290, 296, 299, 302, ...}
Even composite numbers n which are not m-gonal numbers for 3 ≤ m < n
The even numbersn ≥ 4 |
m |
3 ≤ m < n |
n |
m |
k = 3 |
m =
|
The even composite numbers
n ≥ 10 |
n = 6 j + 4 |
j ≥ 1 |
m |
k = 4 |
m = j + 2 |
n ≥ 4 |
m |
3 ≤ m < n |
A274968 Even composite numbers
n |
(n) = n |
n ≥ 4 |
m |
3 ≤ m < n |
- {4, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 266, 272, 278, 284, 290, 296, 302, ...}
Odd composite numbers n which are not m-gonal numbers for 3 ≤ m < n
The odd composite numbersn |
m |
3 ≤ m < n |
- composite numbers
which are divisible by 3 aren
-gonal numbers of order 3, withm
;m = n + 3 3 - odd composite numbers
which are divisible by 5 aren
-gonal numbers of order 5, withm
.m = n + 15 10
n |
(n) = n |
n |
m |
3 ≤ m < n |
- {77, 119, 143, 161, 187, 203, 209, 221, 299, 319, 323, 329, 371, 377, 391, 407, 413, 437, 473, 493, 497, 517, 527, 533, 539, 551, 581, 583, 589, 611, 623, 629, 649, 667, 689, 707, 713, 731, 737, 749, 767, 779, 791, 799, 803, 817, 851, 869, 893, 899, 901, 913, ...}
Sequences
A177025 Number of ways to representn ≥ 3 |
- {1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 1, 3, 2, 1, 2, 2, 1, 2, 3, 1, 2, 1, 1, 2, 2, 2, 4, 1, 1, 2, 2, 1, 2, 1, 1, 4, 2, 1, 2, 2, 1, 3, 2, 1, 2, 3, 1, 2, 2, 1, 2, 1, 1, 2, 3, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 3, 2, 1, 3, 1, 1, 4, 2, 1, 2, 2, 1, 2, 2, 1, 2, 3, 2, 2, 2, 2, 3, 1, 1, 2, 3, ...}
a (n) |
n |
- {3, 6, 15, 36, 225, 561, 1225, 11935, 11781, 27405, 220780, 203841, 3368925, 4921840, 7316001, 33631521, 142629201, 879207616, 1383958576, 3800798001, 12524486976, 181285005825, 118037679760, 239764947345, 738541591425, 1289707733601, 1559439365121, ...}
n |
m ≥ 3 |
n, n ≥ 3, |
m |
- {3, 4, 5, 3, 7, 8, 4, 3, 11, 5, 13, 14, 3, 4, 17, 7, 19, 20, 3, 5, 23, 9, 4, 26, 10, 3, 29, 11, 31, 32, 12, 7, 5, 3, 37, 38, 14, 8, 41, 15, 43, 44, 3, 9, 47, 17, 4, 50, 5, 10, 53, 19, 3, 56, 20, 11, 59, 21, 61, 62, 22, 4, 8, 3, 67, 68, 24, 5, 71, 25, 73, 74, 9, 14, 77, 3, 79, 80, 4, 15, 83, ...}
a (n) |
x |
(x) = n, n ≥ 3; a (n) = 0 |
- {3, 4, 5, 0, 7, 8, 24, 27, 11, 33, 13, 14, 42, 88, 17, 165, 19, 20, 60, 63, 23, 69, 72, 26, 255, 160, 29, 87, 31, 32, 315, 99, 102, 208, 37, 38, 114, 805, 41, 123, 43, 44, 132, 268, 47, 696, 475, 50, 150, 304, 53, 159, 162, 56, 168, 340, 59, 177, 61, 62, 615, 1309, 192, 388, ...}
n, n ≥ 3, |
m |
m |
(n) |
- {2, 2, 2, 3, 2, 2, 3, 4, 2, 3, 2, 2, 5, 4, 2, 3, 2, 2, 6, 4, 2, 3, 5, 2, 3, 7, 2, 3, 2, 2, 3, 4, 5, 8, 2, 2, 3, 4, 2, 3, 2, 2, 9, 4, 2, 3, 7, 2, 6, 4, 2, 3, 10, 2, 3, 4, 2, 3, 2, 2, 3, 8, 5, 11, 2, 2, 3, 7, 2, 3, 2, 2, 5, 4, 2, 12, 2, 2, 9, 4, 2, 3, 5, 2, 3, 4, 2, 3, 13, 8, 3, 4, 5, 6, 2, 2, 3, 10, 2, 3, 2, 2, ...}
A090466 Regular figurative or polygonal numbers of order greater than 2. (Composite numbers which are nontrivial polygonal numbers.)
- {6, 9, 10, 12, 15, 16, 18, 21, 22, 24, 25, 27, 28, 30, 33, 34, 35, 36, 39, 40, 42, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 63, 64, 65, 66, 69, 70, 72, 75, 76, 78, 81, 82, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 99, 100, 102, 105, 106, 108, 111, 112, 114, 115, 117, 118, ...}
1 + k
|
n ≥ 2 |
k ≥ 2 |
- {1, 2, 3, 4, 5, 7, 8, 11, 13, 14, 17, 19, 20, 23, 26, 29, 31, 32, 37, 38, 41, 43, 44, 47, 50, 53, 56, 59, 61, 62, 67, 68, 71, 73, 74, 77, 79, 80, 83, 86, 89, 97, 98, 101, 103, 104, 107, 109, 110, 113, 116, 119, 122, 127, 128, 131, 134, 137, 139, 140, 143, 146, 149, 151, 152, ...}
Polygonal roots
Thek |
rk |
n |
hence
yielding the quadratic roots
n |
n |
n |
k |
n ∈ ℂ |
r3 (10) =
|
r5 (12) =
|
Oblong roots
Since oblong numbers are twice the triangular numbers, we have (by replacing2 n |
n |
i.e.
yielding the oblong roots
r =
|
Schläfli–Poincaré (convex) polytope formula
Schläfli–Poincaré generalization of the Descartes–Euler (convex) polyhedral formula.[3]
For nondegenerate 2-dimensional regular convex polygons:
N0 |
V |
N1 |
E |
Recurrence equation
with initial conditions
Generating function
Order of basis
The order of basis ofN0 |
g |
k n + 1, k > 0, |
k |
{0, 1, …, k − 1} |
1 (mod k) |
k = 5 |
- numbers of form
are expressible as 1 term of the form5 n + 1
;5n + 1 - numbers of form
are expressible as the sum of 2 terms of the form5 n + 2
;5 n + 1 - numbers of form
are expressible as the sum of 3 terms of the form5 n + 3
;5 n + 1 - numbers of form
are expressible as the sum of 4 terms of the form5 n + 4
;5 n + 1 - numbers of form
are expressible as the sum of 5 terms of the form5 n + 0
.5 n + 1
k |
k |
k |
k |
A |
g |
g |
g |
A |
{n 2 | n = 0, 1, 2, …} |
k ≥ 3 |
{P (k, n) | n = 0, 1, 2, …} |
k |
k |
k |
k |
g (d) |
g (d) |
d |
{n d | n = 0, 1, 2, …} |
d |
g (d) |
g (d) |
d ≥ 2 |
In 1997, Conway et al. proved a theorem, called the fifteen theorem,[7] which states that, if a positive definite quadratic form with integer matrix entries represents all natural numbers up to 15, then it represents all natural numbers. This theorem contains Lagrange’s four-square theorem, since every number up to 15 is the sum of at most four squares.
Differences
P (1) N0 − 1 (n) |
n |
N0 |
Partial sums
Tm |
m |
Y (3) N0 + 1 (m) |
m |
N0 |
Partial sums of reciprocals
ForN0 ≠ 4 |
Hm |
m |
γ |
ψ(x) |
For
N0 = 4 |
Sum of reciprocals
ForN0 ≠ 4 |
N0 = 4 |
|
p =
|
x |
x |
y |
x y |
Table of formulae and values
Polygonal numbers associated with constructible polygons (with straightedge and compass) (A003401) are named in bold.
|
Name |
|
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | A-numbers | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3 | Triangular |
|
0 | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 | 66 | 78 | A000217 | |||||
4 | Square |
|
0 | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | A000290 | |||||
5 | Pentagonal |
|
0 | 1 | 5 | 12 | 22 | 35 | 51 | 70 | 92 | 117 | 145 | 176 | 210 | A000326 | |||||
6 | Hexagonal |
|
0 | 1 | 6 | 15 | 28 | 45 | 66 | 91 | 120 | 153 | 190 | 231 | 276 | A000384 | |||||
7 | Heptagonal |
|
0 | 1 | 7 | 18 | 34 | 55 | 81 | 112 | 148 | 189 | 235 | 286 | 342 | A000566 | |||||
8 | Octagonal |
|
0 | 1 | 8 | 21 | 40 | 65 | 96 | 133 | 176 | 225 | 280 | 341 | 408 | A000567 | |||||
9 | 9-gonal |
|
0 | 1 | 9 | 24 | 46 | 75 | 111 | 154 | 204 | 261 | 325 | 396 | 474 | A001106 | |||||
10 | 10-gonal |
|
0 | 1 | 10 | 27 | 52 | 85 | 126 | 175 | 232 | 297 | 370 | 451 | 540 | A001107 | |||||
11 | 11-gonal |
|
0 | 1 | 11 | 30 | 58 | 95 | 141 | 196 | 260 | 333 | 415 | 506 | 606 | A051682 | |||||
12 | 12-gonal |
|
0 | 1 | 12 | 33 | 64 | 105 | 156 | 217 | 288 | 369 | 460 | 561 | 672 | A051624 | |||||
13 | 13-gonal |
|
0 | 1 | 13 | 36 | 70 | 115 | 171 | 238 | 316 | 405 | 505 | 616 | 738 | A051865 | |||||
14 | 14-gonal |
|
0 | 1 | 14 | 39 | 76 | 125 | 186 | 259 | 344 | 441 | 550 | 671 | 804 | A051866 | |||||
15 | 15-gonal |
|
0 | 1 | 15 | 42 | 82 | 135 | 201 | 280 | 372 | 477 | 595 | 726 | 870 | A051867 | |||||
16 | 16-gonal |
|
0 | 1 | 16 | 45 | 88 | 145 | 216 | 301 | 400 | 513 | 640 | 781 | 936 | A051868 | |||||
17 | 17-gonal |
|
0 | 1 | 17 | 48 | 94 | 155 | 231 | 322 | 428 | 549 | 685 | 836 | 1002 | A051869 | |||||
18 | 18-gonal |
|
0 | 1 | 18 | 51 | 100 | 165 | 246 | 343 | 456 | 585 | 730 | 891 | 1068 | A051870 | |||||
19 | 19-gonal |
|
0 | 1 | 19 | 54 | 106 | 175 | 261 | 364 | 484 | 621 | 775 | 946 | 1134 | A051871 | |||||
20 | 20-gonal |
|
0 | 1 | 20 | 57 | 112 | 185 | 276 | 385 | 512 | 657 | 820 | 1001 | 1200 | A051872 | |||||
21 | 21-gonal |
|
0 | 1 | 21 | 60 | 118 | 195 | 291 | 406 | 540 | 693 | 865 | 1056 | 1266 | A051873 | |||||
22 | 22-gonal |
|
0 | 1 | 22 | 63 | 124 | 205 | 306 | 427 | 568 | 729 | 910 | 1111 | 1332 | A051874 | |||||
23 | 23-gonal |
|
0 | 1 | 23 | 66 | 130 | 215 | 321 | 448 | 596 | 765 | 955 | 1166 | 1398 | A051875 | |||||
24 | 24-gonal |
|
0 | 1 | 24 | 69 | 136 | 225 | 336 | 469 | 624 | 801 | 1000 | 1221 | 1464 | A051876 | |||||
25 | 25-gonal |
|
0 | 1 | 25 | 72 | 142 | 235 | 351 | 490 | 652 | 837 | 1045 | 1276 | 1530 | A255184 | |||||
26 | 26-gonal |
|
0 | 1 | 26 | 75 | 148 | 245 | 366 | 511 | 680 | 873 | 1090 | 1331 | 1596 | A255185 | |||||
27 | 27-gonal |
|
0 | 1 | 27 | 78 | 154 | 255 | 381 | 532 | 708 | 909 | 1135 | 1386 | 1662 | A255186 | |||||
28 | 28-gonal |
|
0 | 1 | 28 | 81 | 160 | 265 | 396 | 553 | 736 | 945 | 1180 | 1441 | 1728 | A161935
| |||||
29 | 29-gonal |
|
0 | 1 | 29 | 84 | 166 | 275 | 411 | 574 | 764 | 981 | 1225 | 1496 | 1794 | A255187 | |||||
30 | 30-gonal |
|
0 | 1 | 30 | 87 | 172 | 285 | 426 | 595 | 792 | 1017 | 1270 | 1551 | 1860 | A254474 |
N0 |
N1 |
N0 |
N1 |
Polygonal numbers associated with constructible polygons (with straightedge and compass) (see A003401) are named in bold.
|
Name
Schläfli |
Generating function
|
Order of basis
|
Differences Gnomonic numbers
|
Partial sums
|
Partial sums of reciprocals
|
Sum of Reciprocals[15][16]
| ||||||||||||||||||||||||||||||||||||||||||||||||||
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3 | Triangular
(3, 3) {3} |
3 |
|
| |||||||||||||||||||||||||||||||||||||||||||||||||||||
4 | Square
(4, 4) {4} |
4 |
|
[17] | [18]
Base 10: A013661 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
5 | Pentagonal
(5, 5) {5} |
5 |
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||
6 | Hexagonal
(6, 6) {6} |
6 |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
7 | Heptagonal
(7, 7) {7} |
7 |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
8 | Octagonal
(8, 8) {8} |
8 |
|
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||
9 | 9-gonal
(9, 9) {9} |
9 |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
10 | 10-gonal
(10, 10) {10} |
10 |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
11 | 11-gonal
(11, 11) {11} |
11 |
|
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||
12 | 12-gonal
(12, 12) {12} |
12 |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
13 | 13-gonal
(13, 13) {13} |
13 |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
14 | 14-gonal
(14, 14) {14} |
14 |
|
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||
15 | 15-gonal
(15, 15) {15} |
15 |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
16 | 16-gonal
(16, 16) {16} |
16 |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
17 | 17-gonal
(17, 17) {17} |
17 |
|
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||
18 | 18-gonal
(18, 18) {18} |
18 |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
19 | 19-gonal
(19, 19) {19} |
19 |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
20 | 20-gonal
(20, 20) {20} |
20 |
|
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||
21 | 21-gonal
(21, 21) {21} |
21 |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
22 | 22-gonal
(22, 22) {22} |
22 |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
23 | 23-gonal
(23, 23) {23} |
23 |
|
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||
24 | 24-gonal
(24, 24) {24} |
24 |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
25 | 25-gonal
(25, 25) {25} |
25 |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
26 | 26-gonal
(26, 26) {26} |
26 |
|
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||
27 | 27-gonal
(27, 27) {27} |
27 |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
28 | 28-gonal
(28, 28) {28} |
28 |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||
29 | 29-gonal
(29, 29) {29} |
29 |
|
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||
30 | 30-gonal
(30, 30) {30} |
30 |
|
Table of sequences
See also table of formulae and values above.
|
|
A-numbers | ||
---|---|---|---|---|
3 | 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, ... | A000217 | ||
4 | 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, ... | A000290 | ||
5 | 0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, ... | A000326 | ||
6 | 0, 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946, 1035, 1128, 1225, 1326, 1431, 1540, 1653, 1770, ... | A000384 | ||
7 | 0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, 1918, 2059, ... | A000566 | ||
8 | 0, 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936, 1045, 1160, 1281, 1408, 1541, 1680, 1825, 1976, 2133, 2296, 2465, ... | A000567 | ||
9 | 0, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, 559, 651, 750, 856, 969, 1089, 1216, 1350, 1491, 1639, 1794, 1956, 2125, 2301, 2484, 2674, 2871, ... | A001106 | ||
10 | 0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, ... | A001107 | ||
11 | 0, 1, 11, 30, 58, 95, 141, 196, 260, 333, 415, 506, 606, 715, 833, 960, 1096, 1241, 1395, 1558, 1730, 1911, 2101, 2300, 2508, 2725, 2951, 3186, 3430, ... | A051682 | ||
12 | 0, 1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, 672, 793, 924, 1065, 1216, 1377, 1548, 1729, 1920, 2121, 2332, 2553, 2784, 3025, 3276, 3537, 3808, ... | A051624 | ||
13 | 0, 1, 13, 36, 70, 115, 171, 238, 316, 405, 505, 616, 738, 871, 1015, 1170, 1336, 1513, 1701, 1900, 2110, 2331, 2563, 2806, 3060, 3325, 3601, 3888, 4186, ... | A051865 | ||
14 | 0, 1, 14, 39, 76, 125, 186, 259, 344, 441, 550, 671, 804, 949, 1106, 1275, 1456, 1649, 1854, 2071, 2300, 2541, 2794, 3059, 3336, 3625, 3926, 4239, 4564, ... | A051866 | ||
15 | 0, 1, 15, 42, 82, 135, 201, 280, 372, 477, 595, 726, 870, 1027, 1197, 1380, 1576, 1785, 2007, 2242, 2490, 2751, 3025, 3312, 3612, 3925, 4251, 4590, ... | A051867 | ||
16 | 0, 1, 16, 45, 88, 145, 216, 301, 400, 513, 640, 781, 936, 1105, 1288, 1485, 1696, 1921, 2160, 2413, 2680, 2961, 3256, 3565, 3888, 4225, 4576, 4941, ... | A051868 | ||
17 | 0, 1, 17, 48, 94, 155, 231, 322, 428, 549, 685, 836, 1002, 1183, 1379, 1590, 1816, 2057, 2313, 2584, 2870, 3171, 3487, 3818, 4164, 4525, 4901, 5292, ... | A051869 | ||
18 | 0, 1, 18, 51, 100, 165, 246, 343, 456, 585, 730, 891, 1068, 1261, 1470, 1695, 1936, 2193, 2466, 2755, 3060, 3381, 3718, 4071, 4440, 4825, 5226, 5643, ... | A051870 | ||
19 | 0, 1, 19, 54, 106, 175, 261, 364, 484, 621, 775, 946, 1134, 1339, 1561, 1800, 2056, 2329, 2619, 2926, 3250, 3591, 3949, 4324, 4716, 5125, 5551, 5994, ... | A051871 | ||
20 | 0, 1, 20, 57, 112, 185, 276, 385, 512, 657, 820, 1001, 1200, 1417, 1652, 1905, 2176, 2465, 2772, 3097, 3440, 3801, 4180, 4577, 4992, 5425, 5876, 6345, ... | A051872 | ||
21 | 0, 1, 21, 60, 118, 195, 291, 406, 540, 693, 865, 1056, 1266, 1495, 1743, 2010, 2296, 2601, 2925, 3268, 3630, 4011, 4411, 4830, 5268, 5725, 6201, 6696, ... | A051873 | ||
22 | 0, 1, 22, 63, 124, 205, 306, 427, 568, 729, 910, 1111, 1332, 1573, 1834, 2115, 2416, 2737, 3078, 3439, 3820, 4221, 4642, 5083, 5544, 6025, 6526, 7047, ... | A051874 | ||
23 | 0, 1, 23, 66, 130, 215, 321, 448, 596, 765, 955, 1166, 1398, 1651, 1925, 2220, 2536, 2873, 3231, 3610, 4010, 4431, 4873, 5336, 5820, 6325, 6851, 7398, ... | A051875 | ||
24 | 0, 1, 24, 69, 136, 225, 336, 469, 624, 801, 1000, 1221, 1464, 1729, 2016, 2325, 2656, 3009, 3384, 3781, 4200, 4641, 5104, 5589, 6096, 6625, 7176, 7749, ... | A051876 | ||
25 | 0, 1, 25, 72, 142, 235, 351, 490, 652, 837, 1045, 1276, 1530, 1807, 2107, 2430, 2776, 3145, 3537, 3952, 4390, 4851, 5335, 5842, 6372, 6925, 7501, 8100, ... | A255184 | ||
26 | 0, 1, 26, 75, 148, 245, 366, 511, 680, 873, 1090, 1331, 1596, 1885, 2198, 2535, 2896, 3281, 3690, 4123, 4580, 5061, 5566, 6095, 6648, 7225, 7826, 8451, ... | A255185 | ||
27 | 0, 1, 27, 78, 154, 255, 381, 532, 708, 909, 1135, 1386, 1662, 1963, 2289, 2640, 3016, 3417, 3843, 4294, 4770, 5271, 5797, 6348, 6924, 7525, 8151, 8802, ... | A255186 | ||
28 | 0, 1, 28, 81, 160, 265, 396, 553, 736, 945, 1180, 1441, 1728, 2041, 2380, 2745, 3136, 3553, 3996, 4465, 4960, 5481, 6028, 6601, 7200, 7825, 8476, 9153, ... | A161935
| ||
29 | 0, 1, 29, 84, 166, 275, 411, 574, 764, 981, 1225, 1496, 1794, 2119, 2471, 2850, 3256, 3689, 4149, 4636, 5150, 5691, 6259, 6854, 7476, 8125, 8801, 9504, ... | A255187 | ||
30 | 0, 1, 30, 87, 172, 285, 426, 595, 792, 1017, 1270, 1551, 1860, 2197, 2562, 2955, 3376, 3825, 4302, 4807, 5340, 5901, 6490, 7107, 7752, 8425, 9126, 9855, ... | A254474 |
Mysterious large overlap of terms of A176949 with terms in A140164
A176949: 4, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 77, 80, 86, 98, 104, 110, 116, 119, 122, 128, 134, 140, 143, 146, 152, 158, 161, 164, 170, 182, 187, 188, 194, 200, 203, 206, 209, 212, 218, 221, 224, 230, 236, 242, 248, 254, 266, 272, 278, 284, 290, 296, 299, 302, ... A140164: 1, 2, 4, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 278, 284, 290, 296, 302, ...
See also
Notes
- ↑ Author of the plots: Stefan Friedrich Birkner, License: Creative Commons Attribution-ShareAlike 3.0 Unported.
- ↑ Where
is theP (d ) N0 (n)
-dimensional regular convex polytope number withd
0-dimensional facets, i.e. verticesN0
.V - ↑ Weisstein, Eric W., Polyhedral Formula, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Lagrange's Four-Square Theorem, from MathWorld—A Wolfram Web Resource.
- ↑ 5.0 5.1 Weisstein, Eric W., Fermat's Polygonal Number Theorem, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Waring's Problem, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Fifteen Theorem, from MathWorld—A Wolfram Web Resource.
- ↑ Where
is theY (d ) [(k + 2) + (d − 2)] (n) = Y (d ) k + d (n), k ≥ 1, n ≥ 0,
-dimensional,d
,d ≥ 0
-gonal base (hyper)pyramidal number where, for(k + 2)
, Template:Math is the number of vertices (including the apex vertices) of the polygonal base (hyper)pyramid.d ≥ 2 - ↑ Template:MathWorld
- ↑ Template:MathWorld
- ↑ Template:MathWorld
- ↑ Template:MathWorld
- ↑ Template:MathWorld
- ↑ Template:MathWorld
- ↑ Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
- ↑ Psychedelic Geometry, Inverse Polygonal Numbers Series.
- ↑ Gi-Sang Cheon and Moawwad E. A. El-Mikkawy, Generalized Harmonic Number Identities and a Related Matrix Representation, J. Korean Math. Soc. 44 (2007), No. 2, pp. 487-498.
- ↑ Template:MathWorld
External links
- S. Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
- S. Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
- Herbert S. Wilf, generatingfunctionology, 1994.