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Gnomonic numbers
The gnomonic numbers are zero followed by the arithmetic progressions restricted to a = 1, thus giving restricted to a = 1, which is the only case where all positive integer values of b are coprime to a, thus the b-step gnomonic numbers are given by .
The gnomonic number is the difference between the nth regular convex W-gonal number and the (n-1)th regular convex W-gonal number, where W = b + 2. [1] Gnomonic numbers are frequently assumed to be the square gnomonic numbers, since the original (square) gnomonic numbers where named after the shape corresponding to the differences between two succesive squares.[2]
All figurate numbers are accessible via this structured menu: Classifications of figurate numbers
Contents
- 1 Formulae
- 2 Schläfli-Poincaré (convex) polytope formula
- 3 Recurrence equation
- 4 Generating function
- 5 Order of basis
- 6 Differences
- 7 Partial sums
- 8 Partial sums of reciprocals
- 9 Sum of reciprocals
- 10 Table of formulae and values
- 11 Table of related formulae and values
- 12 Table of sequences
- 13 See also
- 14 Notes
- 15 External links
Formulae
The nth b-step, or W-gonal, b = B-1, W = B+1, gnomonic (the number of sides of a polygon being equal to its number of vertices,) number is given by the formula:[3]
where
and is the nth W-gonal number.
The choices of for labelling the gnomonic numbers and are motivated by the patterns of the (1,k)-Pascal triangle and the (k,1)-Pascal triangle.
These choices are also ideal to highlight the symmetry, for :
where
Schläfli-Poincaré (convex) polytope formula
Schläfli-Poincaré generalization of the Descartes-Euler (convex) polyhedral formula.[4]
For 1-dimensional (d = 1) regular convex polygonal gnomons:
where N0 is the number of 0-dimensional elements (vertices V) of the regular convex polygon gnomon, which is always 2.
Recurrence equation
with initial conditions
where
Generating function
where
Order of basis
The order of basis of W-gonal gnomonic numbers is:
where
The order of basis g for numbers of the form is k, since to represent the numbers in the congruence classes by adding numbers congruent to we need as many terms as the class number, for each congruence classes, e.g. for :
- numbers of form are expressible as 1 term of the form ;
- numbers of form are expressible as the sum of 2 terms of the form ;
- numbers of form are expressible as the sum of 3 terms of the form ;
- numbers of form are expressible as the sum of 4 terms of the form ;
- numbers of form are expressible as the sum of 5 terms of the form .
In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, five pentagonal numbers, and k k-polygonal numbers. Fermat claimed to have a proof of this result, although Fermat's proof has never been found. Joseph Louis Lagrange proved the square case (known as the four squares theorem[5]) in 1770 and Gauss proved the triangular case in 1796. In 1813, Cauchy finally proved the horizontal generalization that every nonnegative integer can be written as a sum of k k-gon numbers (known as the polygonal number theorem,[5]) while a vertical (higher dimensional) generalization has also been made (known as the Hilbert-Waring problem.)
A nonempty subset A of nonnegative integers is called a basis of order g if g is the minimum number with the property that every nonnegative integer can be written as a sum of g elements in A. Lagrange’s sum of four squares can be restated as the set of nonnegative squares forms a basis of order 4.
Theorem (Cauchy) For every , the set of k-gon numbers forms a basis of order k, i.e. every nonnegative integer can be written as a sum of k k-gon numbers.
We note that polygonal numbers are two dimensional analogues of squares. Obviously, cubes, fourth powers, fifth powers, ... are higher dimensional analogues of squares. In 1770, Waring stated without proof that every nonnegative integer can be written as a sum of 4 squares, 9 cubes, 19 fourth powers, and so on. In 1909, Hilbert proved that there is a finite number such that every nonnegative integer is a sum of th powers, i.e. the set of th powers forms a basis of order . The Hilbert-Waring problem[6] is concerned with the study of for . This problem was one of the most important research topics in additive number theory in last 90 years, and it is still a very active area of research.
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
where
Partial sums
where
and is the mth triangular number.
Partial sums of reciprocals
where
and is the digamma function.[8] [9]
Sum of reciprocals
where
The infinite series diverges logarithmically, i.e.:
- as
Table of formulae and values
For , we have:
where
Polygonal gnomon numbers associated with constructible polygons (with straightedge and compass) are named in bold.
B | Name | Formulae
|
n = 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | OEIS
number |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 | Trigonal gnomons | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | A001477(n) | |
3 | Tetragonal gnomons | 0 | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | A004273(n)
A005408(n-1) | |
4 | Pentagonal gnomons | 0 | 1 | 4 | 7 | 10 | 13 | 16 | 19 | 22 | 25 | 28 | 31 | 34 | A016777(n-1) | |
5 | Hexagonal gnomons | 0 | 1 | 5 | 9 | 13 | 17 | 21 | 25 | 29 | 33 | 37 | 41 | 45 | A016813(n-1) | |
6 | Heptagonal gnomons | 0 | 1 | 6 | 11 | 16 | 21 | 26 | 31 | 36 | 41 | 46 | 51 | 56 | A016861(n-1) | |
7 | Octagonal gnomons | 0 | 1 | 7 | 13 | 19 | 25 | 31 | 37 | 43 | 49 | 55 | 61 | 67 | A016921(n-1) | |
8 | Nonagonal gnomons | 0 | 1 | 8 | 15 | 22 | 29 | 36 | 43 | 50 | 57 | 64 | 71 | 78 | A016993(n-1) | |
9 | Decagonal gnomons | 0 | 1 | 9 | 17 | 25 | 33 | 41 | 49 | 57 | 65 | 73 | 81 | 89 | A017077(n-1) | |
10 | Hendecagonal gnomons | 0 | 1 | 10 | 19 | 28 | 37 | 46 | 55 | 64 | 73 | 82 | 91 | 100 | A017173(n-1) | |
11 | Dodecagonal gnomons | 0 | 1 | 11 | 21 | 31 | 41 | 51 | 61 | 71 | 81 | 91 | 101 | 111 | A017281(n-1) | |
12 | Tridecagonal gnomons | 0 | 1 | 12 | 23 | 34 | 45 | 56 | 67 | 78 | 89 | 100 | 111 | 122 | A017401(n-1) | |
13 | Tetradecagonal gnomons | 0 | 1 | 13 | 25 | 37 | 49 | 61 | 73 | 85 | 97 | 109 | 121 | 133 | A017533(n-1) | |
14 | Pentadecagonal gnomons | 0 | 1 | 14 | 27 | 40 | 53 | 66 | 79 | 92 | 105 | 118 | 131 | 144 | A?????? | |
15 | Hexadecagonal gnomons | 0 | 1 | 15 | 29 | 43 | 57 | 71 | 85 | 99 | 113 | 127 | 141 | 155 | A?????? | |
16 | Heptadecagonal gnomons | 0 | 1 | 16 | 31 | 46 | 61 | 76 | 91 | 106 | 121 | 136 | 151 | 166 | A?????? | |
17 | Octadecagonal gnomons | 0 | 1 | 17 | 33 | 49 | 65 | 81 | 97 | 113 | 129 | 145 | 161 | 177 | A?????? | |
18 | Nonadecagonal gnomons | 0 | 1 | 18 | 35 | 52 | 69 | 86 | 103 | 120 | 137 | 154 | 171 | 188 | A?????? | |
19 | Icosagonal gnomons | 0 | 1 | 19 | 37 | 55 | 73 | 91 | 109 | 127 | 145 | 163 | 181 | 199 | A?????? | |
20 | Icosihenagonal gnomons | 0 | 1 | 20 | 39 | 58 | 77 | 96 | 115 | 134 | 153 | 172 | 191 | 210 | A?????? | |
21 | Icosidigonal gnomons | 0 | 1 | 21 | 41 | 61 | 81 | 101 | 121 | 141 | 161 | 181 | 201 | 221 | A?????? | |
22 | Icositrigonal gnomons | 0 | 1 | 22 | 43 | 64 | 85 | 106 | 127 | 148 | 169 | 190 | 211 | 232 | A?????? | |
23 | Icositetragonal gnomons | 0 | 1 | 23 | 45 | 67 | 89 | 111 | 133 | 155 | 177 | 199 | 221 | 243 | A?????? | |
24 | Icosipentagonal gnomons | 0 | 1 | 24 | 47 | 70 | 93 | 116 | 139 | 162 | 185 | 208 | 231 | 254 | A?????? | |
25 | Icosihexagonal gnomons | 0 | 1 | 25 | 49 | 73 | 97 | 121 | 145 | 169 | 193 | 217 | 241 | 265 | A?????? | |
26 | Icosiheptagonal gnomons | 0 | 1 | 26 | 51 | 76 | 101 | 126 | 151 | 176 | 201 | 226 | 251 | 276 | A?????? | |
27 | Icosioctagonal gnomons | 0 | 1 | 27 | 53 | 79 | 105 | 131 | 157 | 183 | 209 | 235 | 261 | 287 | A?????? | |
28 | Icosinonagonal gnomons | 0 | 1 | 28 | 55 | 82 | 109 | 136 | 163 | 190 | 217 | 244 | 271 | 298 | A?????? | |
29 | Triacontagonal gnomons | 0 | 1 | 29 | 57 | 85 | 113 | 141 | 169 | 197 | 225 | 253 | 281 | 309 | A?????? |
Polygonal gnomon numbers associated with constructible polygons (with straightedge and compass) are named in bold.
B | Name | Generating
function
|
Order
of basis[5]
|
Differences
|
Partial sums
|
Partial sums of reciprocals
|
Sum of Reciprocals[10][11]
|
---|---|---|---|---|---|---|---|
2 | Trigonal gnomons |
|
[8] | ||||
3 | Tetragonal gnomons |
|
[8]
|
||||
4 | Pentagonal gnomons |
|
|||||
5 | Hexagonal gnomons |
|
|||||
6 | Heptagonal gnomons |
|
|||||
7 | Octagonal gnomons |
|
|||||
8 | Nonagonal gnomons |
|
|||||
9 | Decagonal gnomons |
|
|||||
10 | Hendecagonal gnomons |
|
|||||
11 | Dodecagonal gnomons |
|
|||||
12 | Tridecagonal gnomons |
|
|||||
13 | Tetradecagonal gnomons |
|
|||||
14 | Pentadecagonal gnomons |
|
|||||
15 | Hexadecagonal gnomons |
|
|||||
16 | Heptadecagonal gnomons |
|
|||||
17 | Octadecagonal gnomons |
|
|||||
18 | Nonadecagonal gnomons |
|
|||||
19 | Icosagonal gnomons |
|
|||||
20 | Icosihenagonal gnomons |
|
|||||
21 | Icosidigonal gnomons |
|
|||||
22 | Icositrigonal gnomons |
|
|||||
23 | Icositetragonal gnomons |
|
|||||
24 | Icosipentagonal gnomons |
|
|||||
25 | Icosihexagonal gnomons |
|
|||||
26 | Icosiheptagonal gnomons |
|
|||||
27 | Icosioctagonal gnomons |
|
|||||
28 | Icosinonagonal gnomons |
|
|||||
29 | Triacontagonal gnomons |
|
Table of sequences
B | sequences |
---|---|
2 | {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, ...} |
3 | {0, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, ...} |
4 | {0, 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, ...} |
5 | {0, 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, ...} |
6 | {0, 1, 6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, 81, 86, 91, 96, 101, 106, 111, 116, 121, 126, 131, 136, 141, 146, 151, 156, 161, 166, 171, 176, 181, ...} |
7 | {0, 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175, 181, 187, 193, 199, 205, 211, ...} |
8 | {0, 1, 8, 15, 22, 29, 36, 43, 50, 57, 64, 71, 78, 85, 92, 99, 106, 113, 120, 127, 134, 141, 148, 155, 162, 169, 176, 183, 190, 197, 204, 211, 218, 225, 232, 239, 246, ...} |
9 | {0, 1, 9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 89, 97, 105, 113, 121, 129, 137, 145, 153, 161, 169, 177, 185, 193, 201, 209, 217, 225, 233, 241, 249, 257, 265, 273, 281, ...} |
10 | {0, 1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 199, 208, 217, 226, 235, 244, 253, 262, 271, 280, 289, 298, 307, ...} |
11 | {0, 1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 201, 211, 221, 231, 241, 251, 261, 271, 281, 291, 301, 311, 321, 331, 341, ...} |
12 | {0, 1, 12, 23, 34, 45, 56, 67, 78, 89, 100, 111, 122, 133, 144, 155, 166, 177, 188, 199, 210, 221, 232, 243, 254, 265, 276, 287, 298, 309, 320, 331, 342, 353, 364, 375, ...} |
13 | {0, 1, 13, 25, 37, 49, 61, 73, 85, 97, 109, 121, 133, 145, 157, 169, 181, 193, 205, 217, 229, 241, 253, 265, 277, 289, 301, 313, 325, 337, 349, 361, 373, 385, 397, 409, ...} |
14 | {0, 1, 14, 27, 40, 53, 66, 79, 92, 105, 118, 131, 144, 157, 170, 183, 196, 209, 222, 235, 248, 261, 274, 287, 300, 313, 326, 339, 352, 365, 378, 391, 404, 417, 430, ...} |
15 | {0, 1, 15, 29, 43, 57, 71, 85, 99, 113, 127, 141, 155, 169, 183, 197, 211, 225, 239, 253, 267, 281, 295, 309, 323, 337, 351, 365, 379, 393, 407, 421, 435, 449, 463, ...} |
16 | {0, 1, 16, 31, 46, 61, 76, 91, 106, 121, 136, 151, 166, 181, 196, 211, 226, 241, 256, 271, 286, 301, 316, 331, 346, 361, 376, 391, 406, 421, 436, 451, 466, 481, 496, ...} |
17 | {0, 1, 17, 33, 49, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, 257, 273, 289, 305, 321, 337, 353, 369, 385, 401, 417, 433, 449, 465, 481, 497, 513, 529, ...} |
18 | {0, 1, 18, 35, 52, 69, 86, 103, 120, 137, 154, 171, 188, 205, 222, 239, 256, 273, 290, 307, 324, 341, 358, 375, 392, 409, 426, 443, 460, 477, 494, 511, 528, 545, 562, ...} |
19 | {0, 1, 19, 37, 55, 73, 91, 109, 127, 145, 163, 181, 199, 217, 235, 253, 271, 289, 307, 325, 343, 361, 379, 397, 415, 433, 451, 469, 487, 505, 523, 541, 559, 577, 595, ...} |
20 | {0, 1, 20, 39, 58, 77, 96, 115, 134, 153, 172, 191, 210, 229, 248, 267, 286, 305, 324, 343, 362, 381, 400, 419, 438, 457, 476, 495, 514, 533, 552, 571, 590, 609, 628, ...} |
21 | {0, 1, 21, 41, 61, 81, 101, 121, 141, 161, 181, 201, 221, 241, 261, 281, 301, 321, 341, 361, 381, 401, 421, 441, 461, 481, 501, 521, 541, 561, 581, 601, 621, 641, 661, ...} |
22 | {0, 1, 22, 43, 64, 85, 106, 127, 148, 169, 190, 211, 232, 253, 274, 295, 316, 337, 358, 379, 400, 421, 442, 463, 484, 505, 526, 547, 568, 589, 610, 631, 652, 673, 694, ...} |
23 | {0, 1, 23, 45, 67, 89, 111, 133, 155, 177, 199, 221, 243, 265, 287, 309, 331, 353, 375, 397, 419, 441, 463, 485, 507, 529, 551, 573, 595, 617, 639, 661, 683, 705, 727, ...} |
24 | {0, 1, 24, 47, 70, 93, 116, 139, 162, 185, 208, 231, 254, 277, 300, 323, 346, 369, 392, 415, 438, 461, 484, 507, 530, 553, 576, 599, 622, 645, 668, 691, 714, 737, 760, ...} |
25 | {0, 1, 25, 49, 73, 97, 121, 145, 169, 193, 217, 241, 265, 289, 313, 337, 361, 385, 409, 433, 457, 481, 505, 529, 553, 577, 601, 625, 649, 673, 697, 721, 745, 769, 793, ...} |
26 | {0, 1, 26, 51, 76, 101, 126, 151, 176, 201, 226, 251, 276, 301, 326, 351, 376, 401, 426, 451, 476, 501, 526, 551, 576, 601, 626, 651, 676, 701, 726, 751, 776, 801, 826, ...} |
27 | {0, 1, 27, 53, 79, 105, 131, 157, 183, 209, 235, 261, 287, 313, 339, 365, 391, 417, 443, 469, 495, 521, 547, 573, 599, 625, 651, 677, 703, 729, 755, 781, 807, 833, 859, ...} |
28 | {0, 1, 28, 55, 82, 109, 136, 163, 190, 217, 244, 271, 298, 325, 352, 379, 406, 433, 460, 487, 514, 541, 568, 595, 622, 649, 676, 703, 730, 757, 784, 811, 838, 865, 892, ...} |
29 | {0, 1, 29, 57, 85, 113, 141, 169, 197, 225, 253, 281, 309, 337, 365, 393, 421, 449, 477, 505, 533, 561, 589, 617, 645, 673, 701, 729, 757, 785, 813, 841, 869, 897, 925, ...} |
See also
Notes
- ↑ An odd number; one of the terms of an arithmetical series by which polygonal numbers are found. Also called gnomonic number. gnomonic (no-mon'ik), a. ..., The Century dictionary: an encyclopedic lexicon of the English ... - Google Books Result, William Dwight Whitney - 1889 - Reference.
- ↑ Weisstein, Eric W., Gnomonic Number, From MathWorld--A Wolfram Web Resource.
- ↑ Where is the d-dimensional regular convex polytope number with N0 0-dimensional facets, i.e. vertices V, or B = b+1 (instead of the 2 vertices) for 1-dimensional regular polytope numbers.
- ↑ Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.
- ↑ 5.0 5.1 5.2 Weisstein, Eric W., Lagrange's Four-Square Theorem, From MathWorld--A Wolfram Web Resource. Cite error: Invalid
<ref>
tag; name "FermatsPolygonalNumberTheorem" defined multiple times with different content - ↑ Weisstein, Eric W., Waring's Problem, From MathWorld--A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Fifteen Theorem, From MathWorld--A Wolfram Web Resource.
- ↑ 8.0 8.1 8.2 8.3 Weisstein, Eric W., Digamma Function, From MathWorld--A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Polygamma Function, From MathWorld--A Wolfram Web Resource.
- ↑ 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.
- ↑ 12.0 12.1 Weisstein, Eric W., Euler-Mascheroni Constant, From MathWorld--A Wolfram Web Resource.
- ↑ Sondow, Jonathan and Weisstein, Eric W., Harmonic Number, From MathWorld--A Wolfram Web Resource.
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, 2nd ed., 1994.