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Gnomonic numbers

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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

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.

Gnomonic numbers formulae and values
B Name Formulae



n = 0 1 2 3 4 5 6 7 8 9 10 11 12 OEIS

number

2 Trigonal gnomons

Triangular gnomons

Nonnegative integers

0 1 2 3 4 5 6 7 8 9 10 11 12 A001477(n)
3 Tetragonal gnomons

Square gnomons

Zero and odd numbers

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??????


Table of related formulae and values

Polygonal gnomon numbers associated with constructible polygons (with straightedge and compass) are named in bold.

Gnomonic numbers related formulae and values
B Name Generating

function


Order

of basis[5]


Differences


Partial sums



Partial sums of reciprocals


[8]

Sum of Reciprocals[10][11]


2 Trigonal gnomons

Triangular gnomons

[8]

[12]

[13]

3 Tetragonal gnomons

Square gnomons

[8]

[12]

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

Gnomonic numbers 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

Centered gnomonic numbers

Notes

  1. 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.
  2. Weisstein, Eric W., Gnomonic Number, From MathWorld--A Wolfram Web Resource.
  3. 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.
  4. Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.
  5. 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
  6. Weisstein, Eric W., Waring's Problem, From MathWorld--A Wolfram Web Resource.
  7. Weisstein, Eric W., Fifteen Theorem, From MathWorld--A Wolfram Web Resource.
  8. 8.0 8.1 8.2 8.3 Weisstein, Eric W., Digamma Function, From MathWorld--A Wolfram Web Resource.
  9. Weisstein, Eric W., Polygamma Function, From MathWorld--A Wolfram Web Resource.
  10. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
  11. PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES.
  12. 12.0 12.1 Weisstein, Eric W., Euler-Mascheroni Constant, From MathWorld--A Wolfram Web Resource.
  13. Sondow, Jonathan and Weisstein, Eric W., Harmonic Number, From MathWorld--A Wolfram Web Resource.

External links