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Centered gnomonic numbers

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The centered gnomonic numbers are the centered equidistributions of dots, restricted to a = 1, which is the only case where all positive integer values of b are coprime to a, and where b is a positive integer, (Cf. gnomonic numbers,) thus the resulting b-step centered gnomonic numbers are given by:

, when b is odd positive integer,
, when b is even positive integer;

or more compactly:

or even more compactly:

The two cases are explained as follow:

  • when b is even positive integer, is always odd, so when centered, any previous dot is masked by a new dot, thus we get dots as a result;
  • when b is odd positive integer, is alternatively odd then even:
 n = 0:                                      O                                          gives 1 (central "O") dot since 1 is odd,
 n = 1:                   e   ...   e e e e eOe e e e e   ...   e                       gives 1 "O" + (1+b) "e" = 2+b dots since 1+b is even,
 n = 2:          o  ...  oeo  ...  oeoeoeoeoeOeoeoeoeoeo  ...  oeo  ...  o              gives 1 "O" + 2b "o" + (1+b) "e" = 2+3b dots since 1+2b is odd,
 n = 3:   e ... eoe ... eoeoe ... eoeoeoeoeoeOeoeoeoeoeoe ... eoeoe ... eoe ... e       gives 1 "O" + 2b "o" + (1+3b) "e" = 2+5b dots since 1+3b is even,
  ...                                       ...                                                                    ...

so we alternatively mask the dots of a given parity (with 2b more, for n ≥ 2) while the dots of the other parity remain untouched, thus we get dots as a result.


All figurate numbers are accessible via this structured menu: Classifications of figurate numbers

Formulae

The nth b-step, or B = b+1, centered gnomonic number is given by the formula:

where

The choice of for labelling the centered gnomonic numbers is motivated by the patterns of the (1,k)-Pascal triangle and the (k,1)-Pascal triangle, which suggested this choice for the gnomonic numbers.

Schläfli-Poincaré (convex) polytope formula

Schläfli-Poincaré generalization of the Descartes-Euler (convex) polyhedral formula.[1]

For 1-dimensional (d = 1) regular convex polytopes:

where N0 is the number of 0-dimensional elements (vertices V) of the 1-dimensional (d = 1) regular convex polytope, which is always 2.

Recurrence equation

with initial conditions

where

Generating function

where

Order of basis

The order of basis of centered 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[2]) 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,[2]) 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[3] 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,[4] 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

Partial sums of reciprocals

where

Sum of reciprocals

where

The infinite series diverges logarithmically, i.e.:

as

Table of formulae and values

Centered 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


Odd numbers

1 3 5 7 9 11 13 15 17 19 21 23 25 A005408(n)
3


Odd numbers

1 3 5 7 9 11 13 15 17 19 21 23 25 A005408(n)
4 1 5 11 17 23 29 35 41 47 53 59 65 71 A101328(n+2)
5 1 5 9 13 17 21 25 29 33 37 41 45 49 A016813(n)
6 1 7 17 27 37 47 57 67 77 87 97 107 117 A160912(n+1)
7 1 7 13 19 25 31 37 43 49 55 61 67 73 A016921(n)
8 1 9 23 37 51 65 79 93 107 121 135 149 163 Axxxxxx
9 1 9 17 25 33 41 49 57 65 73 81 89 97 A017077(n)
10 1 11 29 47 65 83 101 119 137 155 173 191 209 Axxxxxx
11 1 11 21 31 41 51 61 71 81 91 101 111 121 A017281(n)
12 1 13 35 57 79 101 123 145 167 189 211 233 255 Axxxxxx
13 1 13 25 37 49 61 73 85 97 109 121 133 145 A017533(n)
14 1 15 41 67 93 119 145 171 197 223 249 275 301 Axxxxxx
15 1 15 29 43 57 71 85 99 113 127 141 155 169 Axxxxxx
16 1 17 47 77 107 137 167 197 227 257 287 317 347 Axxxxxx
17 1 17 33 49 65 81 97 113 129 145 161 177 193 Axxxxxx
18 1 19 53 87 121 155 189 223 257 291 325 359 393 Axxxxxx
19 1 19 37 55 73 91 109 127 145 163 181 199 217 Axxxxxx
20 1 21 59 97 135 173 211 249 287 325 363 401 439 Axxxxxx
21 1 21 41 61 81 101 121 141 161 181 201 221 241 Axxxxxx
22 1 23 65 107 149 191 233 275 317 359 401 443 485 Axxxxxx
23 1 23 45 67 89 111 133 155 177 199 221 243 265 Axxxxxx
24 1 25 71 117 163 209 255 301 347 393 439 485 531 Axxxxxx
25 1 25 49 73 97 121 145 169 193 217 241 265 289 Axxxxxx
26 1 27 77 127 177 227 277 327 377 427 477 527 577 Axxxxxx
27 1 27 53 79 105 131 157 183 209 235 261 287 313 Axxxxxx
28 1 29 83 137 191 245 299 353 407 461 515 569 623 Axxxxxx
29 1 29 57 85 113 141 169 197 225 253 281 309 337 Axxxxxx


Table of related formulae and values

Centered gnomonic numbers related formulae and values
B Name Generating

function


Order

of basis[2]

Differences

Partial sums

Partial sums of reciprocals

Sums of reciprocals


2

[5]

[6]

3

[5]

[6]

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


Table of sequences

Centered gnomonic numbers sequences
B sequences
2 {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, ...}
3 {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 {1, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 197, 203, 209, ...}
5 {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 {1, 7, 17, 27, 37, 47, 57, 67, 77, 87, 97, 107, 117, 127, 137, 147, 157, 167, 177, 187, 197, 207, 217, 227, 237, 247, 257, 267, 277, 287, 297, 307, 317, 327, 337, ...}
7 {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 {1, 9, 23, 37, 51, 65, 79, 93, 107, 121, 135, 149, 163, 177, 191, 205, 219, 233, 247, 261, 275, 289, 303, 317, 331, 345, 359, 373, 387, 401, 415, 429, 443, 457, 471, ...}
9 {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 {1, 11, 29, 47, 65, 83, 101, 119, 137, 155, 173, 191, 209, 227, 245, 263, 281, 299, 317, 335, 353, 371, 389, 407, 425, 443, 461, 479, 497, 515, 533, 551, 569, 587, 605, ...}
11 {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 {1, 13, 35, 57, 79, 101, 123, 145, 167, 189, 211, 233, 255, 277, 299, 321, 343, 365, 387, 409, 431, 453, 475, 497, 519, 541, 563, 585, 607, 629, 651, 673, 695, 717, ...}
13 {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 {1, 15, 41, 67, 93, 119, 145, 171, 197, 223, 249, 275, 301, 327, 353, 379, 405, 431, 457, 483, 509, 535, 561, 587, 613, 639, 665, 691, 717, 743, 769, 795, 821, ...}
15 {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 {1, 17, 47, 77, 107, 137, 167, 197, 227, 257, 287, 317, 347, 377, 407, 437, 467, 497, 527, 557, 587, 617, 647, 677, 707, 737, 767, 797, 827, 857, 887, 917, 947, ...}
17 {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 {1, 19, 53, 87, 121, 155, 189, 223, 257, 291, 325, 359, 393, 427, 461, 495, 529, 563, 597, 631, 665, 699, 733, 767, 801, 835, 869, 903, 937, 971, 1005, 1039, 1073, ...}
19 {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 {1, 21, 59, 97, 135, 173, 211, 249, 287, 325, 363, 401, 439, 477, 515, 553, 591, 629, 667, 705, 743, 781, 819, 857, 895, 933, 971, 1009, 1047, 1085, 1123, 1161, ...}
21 {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 {1, 23, 65, 107, 149, 191, 233, 275, 317, 359, 401, 443, 485, 527, 569, 611, 653, 695, 737, 779, 821, 863, 905, 947, 989, 1031, 1073, 1115, 1157, 1199, 1241, 1283, ...}
23 {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 {1, 25, 71, 117, 163, 209, 255, 301, 347, 393, 439, 485, 531, 577, 623, 669, 715, 761, 807, 853, 899, 945, 991, 1037, 1083, 1129, 1175, 1221, 1267, 1313, 1359, 1405, ...}
25 {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 {1, 27, 77, 127, 177, 227, 277, 327, 377, 427, 477, 527, 577, 627, 677, 727, 777, 827, 877, 927, 977, 1027, 1077, 1127, 1177, 1227, 1277, 1327, 1377, 1427, 1477, ...}
27 {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 {1, 29, 83, 137, 191, 245, 299, 353, 407, 461, 515, 569, 623, 677, 731, 785, 839, 893, 947, 1001, 1055, 1109, 1163, 1217, 1271, 1325, 1379, 1433, 1487, 1541, 1595, ...}
29 {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

Gnomonic numbers

Notes

  1. Weisstein, Eric W., Polyhedral Formula, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/PolyhedralFormula.html].
  2. 2.0 2.1 2.2 Weisstein, Eric W., Lagrange's Four-Square Theorem, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/LagrangesFour-SquareTheorem.html]. Cite error: Invalid <ref> tag; name "FermatsPolygonalNumberTheorem" defined multiple times with different content
  3. Weisstein, Eric W., Waring's Problem, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/WaringsProblem.html].
  4. Weisstein, Eric W., Fifteen Theorem, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/FifteenTheorem.html].
  5. 5.0 5.1 Weisstein, Eric W., Digamma Function, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/DigammaFunction.html]. Cite error: Invalid <ref> tag; name "DigammaFunction" defined multiple times with different content
  6. 6.0 6.1 Weisstein, Eric W., Euler-Mascheroni Constant, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Euler-MascheroniConstant.html].

External links