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# Gnomonic numbers

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The gnomonic numbers are zero followed by the arithmetic progressions ${\displaystyle \scriptstyle a+(n-1)b,\ (a,b)=1,\ n\geq 1\,}$ restricted to a = 1, thus giving ${\displaystyle \scriptstyle a+(n-1)b-(a-b)0^{n},\ (a,b)=1,\ n\geq 0\,}$ 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 ${\displaystyle \scriptstyle 1+(n-1)b-(1-b)0^{n},\ n\geq 0\,}$.

The gnomonic number ${\displaystyle \scriptstyle 1+(n-1)b-(1-b)0^{n},\ n\geq 0\,}$ 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]

${\displaystyle P_{B}^{(1)}(n)=P_{W}^{(2)}(n)-P_{W}^{(2)}(n-1)=1+(B-1)(n-1)+(B-2)0^{n},\ n\geq 0,\,}$

where

${\displaystyle B=b+1=W-1\geq 2,\,}$

and ${\displaystyle \scriptstyle P_{W}^{(2)}(n)\,}$ is the nth W-gonal number.

The choices of ${\displaystyle \scriptstyle B=b+1=W-1\geq 2\,}$ for labelling the gnomonic numbers and ${\displaystyle \scriptstyle P_{B}^{(1)}(0)=0,\ P_{B}^{(1)}(1)=1\,}$ 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 ${\displaystyle \scriptstyle n\geq 2\,}$:

${\displaystyle P_{B}^{(1)}(n)=P_{n}^{(1)}(B),\,}$

where

${\displaystyle B=b+1=W-1\geq 2.\,}$

## 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:

${\displaystyle {\sum _{i=0}^{d-1}(-1)^{i}N_{i}}=N_{0}=V=2,\,}$

where N0 is the number of 0-dimensional elements (vertices V) of the regular convex polygon gnomon, which is always 2.

## Recurrence equation

${\displaystyle P_{B}^{(1)}(n)=P_{B}^{(1)}(n-1)+(B-1),\ n\geq 2,\,}$

with initial conditions

${\displaystyle P_{B}^{(1)}(0)=0,\,}$
${\displaystyle P_{B}^{(1)}(1)=1,\,}$

where

${\displaystyle B=b+1=W-1\ \geq 2.\,}$

## Generating function

${\displaystyle G_{\{P_{B}^{(1)}(n)\}}(x)={\frac {x\{1+(B-2)x\}}{(1-x)^{2}}},\,}$

where

${\displaystyle B=b+1=W-1\ \geq 2.\,}$

## Order of basis

The order of basis of W-gonal gnomonic numbers is:

${\displaystyle g_{\{P_{B}^{(1)}\}}=B-1,\,}$

where

${\displaystyle B=b+1=W-1\ \geq 2.\,}$

The order of basis g for numbers of the form ${\displaystyle \scriptstyle kn+1,\ k>0\,}$ is k, since to represent the numbers in the congruence classes ${\displaystyle \scriptstyle \{0,1,...,k-1\}\,}$ by adding numbers congruent to ${\displaystyle \scriptstyle 1\mod k\,}$ we need as many terms as the class number, for each congruence classes, e.g. for ${\displaystyle \scriptstyle k=5\,}$:

numbers of form ${\displaystyle \scriptstyle 5n+1\,}$ are expressible as 1 term of the form ${\displaystyle \scriptstyle 5n+1\,}$;
numbers of form ${\displaystyle \scriptstyle 5n+2\,}$ are expressible as the sum of 2 terms of the form ${\displaystyle \scriptstyle 5n+1\,}$;
numbers of form ${\displaystyle \scriptstyle 5n+3\,}$ are expressible as the sum of 3 terms of the form ${\displaystyle \scriptstyle 5n+1\,}$;
numbers of form ${\displaystyle \scriptstyle 5n+4\,}$ are expressible as the sum of 4 terms of the form ${\displaystyle \scriptstyle 5n+1\,}$;
numbers of form ${\displaystyle \scriptstyle 5n+0\,}$ are expressible as the sum of 5 terms of the form ${\displaystyle \scriptstyle 5n+1\,}$.

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 ${\displaystyle \scriptstyle \{n^{2}|n=0,1,2,\ldots \}\,}$ of nonnegative squares forms a basis of order 4.

Theorem (Cauchy) For every ${\displaystyle \scriptstyle k\geq 3}$, the set ${\displaystyle \scriptstyle \{P(k,n)|n=0,1,2,\ldots \}\,}$ 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 ${\displaystyle \scriptstyle g(d)\,}$ such that every nonnegative integer is a sum of ${\displaystyle \scriptstyle g(d)\,}$ ${\displaystyle \scriptstyle d\,}$th powers, i.e. the set ${\displaystyle \scriptstyle \{n^{d}|n=0,1,2,\ldots \}\,}$ of ${\displaystyle \scriptstyle d\,}$th powers forms a basis of order ${\displaystyle \scriptstyle g(d)\,}$. The Hilbert-Waring problem[6] is concerned with the study of ${\displaystyle \scriptstyle g(d)\,}$ for ${\displaystyle \scriptstyle d\geq 2\,}$. 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

${\displaystyle P_{B}^{(1)}(n)-P_{B}^{(1)}(n-1)=0\,\ n=0,}$
${\displaystyle P_{B}^{(1)}(n)-P_{B}^{(1)}(n-1)=1\,\ n=1,}$
${\displaystyle P_{B}^{(1)}(n)-P_{B}^{(1)}(n-1)=B-1,\ n\geq 2,\,}$

where

${\displaystyle B=b+1=W-1\geq 2.\,}$

## Partial sums

${\displaystyle \sum _{n=1}^{m}P_{B}^{(1)}(n)=P_{B+1}^{(2)}(m)=m+(B-1)P_{3}^{(2)}(m-1)=m+(B-1)T_{m-1}=m+(B-1){\binom {m}{2}}}$
${\displaystyle =m+(B-1){(m-1)m \over 2}={\frac {m}{2}}[(B-1)m-(B-3)],\,}$

where

${\displaystyle B=b+1=W-1\geq 2\,}$

and ${\displaystyle \scriptstyle P_{3}^{(2)}(m)=T_{m}\,}$ is the mth triangular number.

## Partial sums of reciprocals

${\displaystyle \sum _{n=1}^{m}{\frac {1}{P_{B}^{(1)}(n)}}={\frac {\psi (m+{\frac {1}{B-1}})-\psi ({\frac {1}{B-1}})}{B-1}},\,}$

where

${\displaystyle B=b+1=W-1\geq 2.\,}$

and ${\displaystyle \scriptstyle \psi (x)\,}$ is the digamma function.[8] [9]

## Sum of reciprocals

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{P_{B}^{(1)}(n)}}=\infty ,\,}$

where

${\displaystyle B=b+1=W-1\geq 2.\,}$

The infinite series diverges logarithmically, i.e.:

${\displaystyle \sum _{n=1}^{m}{\frac {1}{P_{B}^{(1)}(n)}}\sim O(log(m)),\,}$ as ${\displaystyle m\to \infty .\,}$

## Table of formulae and values

For ${\displaystyle \scriptstyle n\geq 2\,}$, we have:

${\displaystyle P_{B}^{(1)}(n)=P_{n}^{(1)}(B),\,}$

where

${\displaystyle B=b+1=W-1\geq 2.\,}$

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

Gnomonic numbers formulae and values
B Name Formulae

${\displaystyle P_{B}^{(1)}(n)=\,}$

${\displaystyle \scriptstyle 1+(B-1)(n-1)+(B-2)\ 0^{n},\,}$

${\displaystyle \scriptstyle B\geq 2\,}$

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

number

2 Trigonal gnomons ${\displaystyle n\,}$ 0 1 2 3 4 5 6 7 8 9 10 11 12 A001477(n)
3 Tetragonal gnomons ${\displaystyle 1+2(n-1)+0^{n}\,}$ 0 1 3 5 7 9 11 13 15 17 19 21 23 A004273(n)

A005408(n-1)

4 Pentagonal gnomons ${\displaystyle 1+3(n-1)+2\ 0^{n}\,}$ 0 1 4 7 10 13 16 19 22 25 28 31 34 A016777(n-1)
5 Hexagonal gnomons ${\displaystyle 1+4(n-1)+3\ 0^{n}\,}$ 0 1 5 9 13 17 21 25 29 33 37 41 45 A016813(n-1)
6 Heptagonal gnomons ${\displaystyle 1+5(n-1)+4\ 0^{n}\,}$ 0 1 6 11 16 21 26 31 36 41 46 51 56 A016861(n-1)
7 Octagonal gnomons ${\displaystyle 1+6(n-1)+5\ 0^{n}\,}$ 0 1 7 13 19 25 31 37 43 49 55 61 67 A016921(n-1)
8 Nonagonal gnomons ${\displaystyle 1+7(n-1)+6\ 0^{n}\,}$ 0 1 8 15 22 29 36 43 50 57 64 71 78 A016993(n-1)
9 Decagonal gnomons ${\displaystyle 1+8(n-1)+7\ 0^{n}\,}$ 0 1 9 17 25 33 41 49 57 65 73 81 89 A017077(n-1)
10 Hendecagonal gnomons ${\displaystyle 1+9(n-1)+8\ 0^{n}\,}$ 0 1 10 19 28 37 46 55 64 73 82 91 100 A017173(n-1)
11 Dodecagonal gnomons ${\displaystyle 1+10(n-1)+9\ 0^{n}\,}$ 0 1 11 21 31 41 51 61 71 81 91 101 111 A017281(n-1)
12 Tridecagonal gnomons ${\displaystyle 1+11(n-1)+10\ 0^{n}\,}$ 0 1 12 23 34 45 56 67 78 89 100 111 122 A017401(n-1)
13 Tetradecagonal gnomons ${\displaystyle 1+12(n-1)+11\ 0^{n}\,}$ 0 1 13 25 37 49 61 73 85 97 109 121 133 A017533(n-1)
14 Pentadecagonal gnomons ${\displaystyle 1+13(n-1)+12\ 0^{n}\,}$ 0 1 14 27 40 53 66 79 92 105 118 131 144 A??????
15 Hexadecagonal gnomons ${\displaystyle 1+14(n-1)+13\ 0^{n}\,}$ 0 1 15 29 43 57 71 85 99 113 127 141 155 A??????
16 Heptadecagonal gnomons ${\displaystyle 1+15(n-1)+14\ 0^{n}\,}$ 0 1 16 31 46 61 76 91 106 121 136 151 166 A??????
17 Octadecagonal gnomons ${\displaystyle 1+16(n-1)+15\ 0^{n}\,}$ 0 1 17 33 49 65 81 97 113 129 145 161 177 A??????
18 Nonadecagonal gnomons ${\displaystyle 1+17(n-1)+16\ 0^{n}\,}$ 0 1 18 35 52 69 86 103 120 137 154 171 188 A??????
19 Icosagonal gnomons ${\displaystyle 1+18(n-1)+17\ 0^{n}\,}$ 0 1 19 37 55 73 91 109 127 145 163 181 199 A??????
20 Icosihenagonal gnomons ${\displaystyle 1+19(n-1)+18\ 0^{n}\,}$ 0 1 20 39 58 77 96 115 134 153 172 191 210 A??????
21 Icosidigonal gnomons ${\displaystyle 1+20(n-1)+19\ 0^{n}\,}$ 0 1 21 41 61 81 101 121 141 161 181 201 221 A??????
22 Icositrigonal gnomons ${\displaystyle 1+21(n-1)+20\ 0^{n}\,}$ 0 1 22 43 64 85 106 127 148 169 190 211 232 A??????
23 Icositetragonal gnomons ${\displaystyle 1+22(n-1)+21\ 0^{n}\,}$ 0 1 23 45 67 89 111 133 155 177 199 221 243 A??????
24 Icosipentagonal gnomons ${\displaystyle 1+23(n-1)+22\ 0^{n}\,}$ 0 1 24 47 70 93 116 139 162 185 208 231 254 A??????
25 Icosihexagonal gnomons ${\displaystyle 1+24(n-1)+23\ 0^{n}\,}$ 0 1 25 49 73 97 121 145 169 193 217 241 265 A??????
26 Icosiheptagonal gnomons ${\displaystyle 1+25(n-1)+24\ 0^{n}\,}$ 0 1 26 51 76 101 126 151 176 201 226 251 276 A??????
27 Icosioctagonal gnomons ${\displaystyle 1+26(n-1)+25\ 0^{n}\,}$ 0 1 27 53 79 105 131 157 183 209 235 261 287 A??????
28 Icosinonagonal gnomons ${\displaystyle 1+27(n-1)+26\ 0^{n}\,}$ 0 1 28 55 82 109 136 163 190 217 244 271 298 A??????
29 Triacontagonal gnomons ${\displaystyle 1+28(n-1)+27\ 0^{n}\,}$ 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

${\displaystyle G_{\{P_{B}^{(1)}(n)\}}(x)=\,}$

${\displaystyle \scriptstyle {\frac {x\{1+(B-2)x\}}{(1-x)^{2}}}\,}$

Order

of basis[5]

${\displaystyle g_{\{P_{B}^{(1)}\}}=\,}$

${\displaystyle \scriptstyle B-1,\ B\geq 2\,}$

Differences

${\displaystyle P_{B}^{(1)}(n)-\,}$

${\displaystyle P_{B}^{(1)}(n-1)=\,}$

${\displaystyle \scriptstyle B-1,\ B\geq 2,\ n\geq 2\,}$

Partial sums

${\displaystyle \sum _{n=1}^{m}P_{B}^{(1)}(n)=\,}$

${\displaystyle P_{B+1}^{(2)}(m)=\,}$

${\displaystyle \scriptstyle m+(B-1){\binom {m}{2}},\ B\geq 2\,}$

Partial sums of reciprocals

${\displaystyle \sum _{n=1}^{m}{1 \over {P_{B}^{(1)}(n)}}=\,}$

${\displaystyle \scriptstyle {\frac {\psi (m+{\frac {1}{B-1}})-\psi ({\frac {1}{B-1}})}{B-1}},\ B\geq 2\,}$ [8]

Sum of Reciprocals[10][11]

${\displaystyle \sum _{n=1}^{\infty }{1 \over {P_{B}^{(1)}(n)}}=}$

${\displaystyle \infty \,}$

2 Trigonal gnomons ${\displaystyle {\frac {x}{(1-x)^{2}}}\,}$ ${\displaystyle 1\,}$ ${\displaystyle 1\,}$ ${\displaystyle P_{3}^{(2)}(m)=\,}$

${\displaystyle m+1{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{1}})-\psi ({\frac {1}{1}})}{1}}\,}$ [8]

${\displaystyle \psi (m+1)+\gamma \,}$ [12]

${\displaystyle H_{m}\,}$ [13]

${\displaystyle \infty \,}$
3 Tetragonal gnomons ${\displaystyle {\frac {x(1+x)}{(1-x)^{2}}}\,}$ ${\displaystyle 2\,}$ ${\displaystyle 2\,}$ ${\displaystyle P_{4}^{(2)}(m)=\,}$

${\displaystyle m+2{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{2}})-\psi ({\frac {1}{2}})}{2}}\,}$ [8]

${\displaystyle {\frac {\psi (m+{\frac {1}{2}})+\gamma +2\log(2)}{2}}\,}$ [12]

${\displaystyle {\frac {H_{(m-{\frac {1}{2}})}}{2}}+log(2)\,}$

${\displaystyle \infty \,}$
4 Pentagonal gnomons ${\displaystyle {\frac {x(1+2x)}{(1-x)^{2}}}\,}$ ${\displaystyle 3\,}$ ${\displaystyle 3\,}$ ${\displaystyle P_{5}^{(2)}(m)=\,}$

${\displaystyle m+3{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{3}})-\psi ({\frac {1}{3}})}{3}}\,}$ ${\displaystyle \infty \,}$
5 Hexagonal gnomons ${\displaystyle {\frac {x(1+3x)}{(1-x)^{2}}}\,}$ ${\displaystyle 4\,}$ ${\displaystyle 4\,}$ ${\displaystyle P_{6}^{(2)}(m)=\,}$

${\displaystyle m+4{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{4}})-\psi ({\frac {1}{4}})}{4}}\,}$ ${\displaystyle \infty \,}$
6 Heptagonal gnomons ${\displaystyle {\frac {x(1+4x)}{(1-x)^{2}}}\,}$ ${\displaystyle 5\,}$ ${\displaystyle 5\,}$ ${\displaystyle P_{7}^{(2)}(m)=\,}$

${\displaystyle m+5{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{5}})-\psi ({\frac {1}{5}})}{5}}\,}$ ${\displaystyle \infty \,}$
7 Octagonal gnomons ${\displaystyle {\frac {x(1+5x)}{(1-x)^{2}}}\,}$ ${\displaystyle 6\,}$ ${\displaystyle 6\,}$ ${\displaystyle P_{8}^{(2)}(m)=\,}$

${\displaystyle m+6{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{6}})-\psi ({\frac {1}{6}})}{6}}\,}$ ${\displaystyle \infty \,}$
8 Nonagonal gnomons ${\displaystyle {\frac {x(1+6x)}{(1-x)^{2}}}\,}$ ${\displaystyle 7\,}$ ${\displaystyle 7\,}$ ${\displaystyle P_{9}^{(2)}(m)=\,}$

${\displaystyle m+7{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{7}})-\psi ({\frac {1}{7}})}{7}}\,}$ ${\displaystyle \infty \,}$
9 Decagonal gnomons ${\displaystyle {\frac {x(1+7x)}{(1-x)^{2}}}\,}$ ${\displaystyle 8\,}$ ${\displaystyle 8\,}$ ${\displaystyle P_{10}^{(2)}(m)=\,}$

${\displaystyle m+8{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{8}})-\psi ({\frac {1}{8}})}{8}}\,}$ ${\displaystyle \infty \,}$
10 Hendecagonal gnomons ${\displaystyle {\frac {x(1+8x)}{(1-x)^{2}}}\,}$ ${\displaystyle 9\,}$ ${\displaystyle 9\,}$ ${\displaystyle P_{11}^{(2)}(m)=\,}$

${\displaystyle m+9{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{9}})-\psi ({\frac {1}{9}})}{9}}\,}$ ${\displaystyle \infty \,}$
11 Dodecagonal gnomons ${\displaystyle {\frac {x(1+9x)}{(1-x)^{2}}}\,}$ ${\displaystyle 10\,}$ ${\displaystyle 10\,}$ ${\displaystyle P_{12}^{(2)}(m)=\,}$

${\displaystyle m+10{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{10}})-\psi ({\frac {1}{10}})}{10}}\,}$ ${\displaystyle \infty \,}$
12 Tridecagonal gnomons ${\displaystyle {\frac {x(1+10x)}{(1-x)^{2}}}\,}$ ${\displaystyle 11\,}$ ${\displaystyle 11\,}$ ${\displaystyle P_{13}^{(2)}(m)=\,}$

${\displaystyle m+11{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{11}})-\psi ({\frac {1}{11}})}{11}}\,}$ ${\displaystyle \infty \,}$
13 Tetradecagonal gnomons ${\displaystyle {\frac {x(1+11x)}{(1-x)^{2}}}\,}$ ${\displaystyle 12\,}$ ${\displaystyle 12\,}$ ${\displaystyle P_{14}^{(2)}(m)=\,}$

${\displaystyle m+12{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{12}})-\psi ({\frac {1}{12}})}{12}}\,}$ ${\displaystyle \infty \,}$
14 Pentadecagonal gnomons ${\displaystyle {\frac {x(1+12x)}{(1-x)^{2}}}\,}$ ${\displaystyle 13\,}$ ${\displaystyle 13\,}$ ${\displaystyle P_{15}^{(2)}(m)=\,}$

${\displaystyle m+13{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{13}})-\psi ({\frac {1}{13}})}{13}}\,}$ ${\displaystyle \infty \,}$
15 Hexadecagonal gnomons ${\displaystyle {\frac {x(1+13x)}{(1-x)^{2}}}\,}$ ${\displaystyle 14\,}$ ${\displaystyle 14\,}$ ${\displaystyle P_{16}^{(2)}(m)=\,}$

${\displaystyle m+14{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{14}})-\psi ({\frac {1}{14}})}{14}}\,}$ ${\displaystyle \infty \,}$
16 Heptadecagonal gnomons ${\displaystyle {\frac {x(1+14x)}{(1-x)^{2}}}\,}$ ${\displaystyle 15\,}$ ${\displaystyle 15\,}$ ${\displaystyle P_{17}^{(2)}(m)=\,}$

${\displaystyle m+15{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{15}})-\psi ({\frac {1}{15}})}{15}}\,}$ ${\displaystyle \infty \,}$
17 Octadecagonal gnomons ${\displaystyle {\frac {x(1+15x)}{(1-x)^{2}}}\,}$ ${\displaystyle 16\,}$ ${\displaystyle 16\,}$ ${\displaystyle P_{18}^{(2)}(m)=\,}$

${\displaystyle m+16{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{16}})-\psi ({\frac {1}{16}})}{16}}\,}$ ${\displaystyle \infty \,}$
18 Nonadecagonal gnomons ${\displaystyle {\frac {x(1+16x)}{(1-x)^{2}}}\,}$ ${\displaystyle 17\,}$ ${\displaystyle 17\,}$ ${\displaystyle P_{19}^{(2)}(m)=\,}$

${\displaystyle m+17{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{17}})-\psi ({\frac {1}{17}})}{17}}\,}$ ${\displaystyle \infty \,}$
19 Icosagonal gnomons ${\displaystyle {\frac {x(1+17x)}{(1-x)^{2}}}\,}$ ${\displaystyle 18\,}$ ${\displaystyle 18\,}$ ${\displaystyle P_{20}^{(2)}(m)=\,}$

${\displaystyle m+18{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{18}})-\psi ({\frac {1}{18}})}{18}}\,}$ ${\displaystyle \infty \,}$
20 Icosihenagonal gnomons ${\displaystyle {\frac {x(1+18x)}{(1-x)^{2}}}\,}$ ${\displaystyle 19\,}$ ${\displaystyle 19\,}$ ${\displaystyle P_{21}^{(2)}(m)=\,}$

${\displaystyle m+19{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{19}})-\psi ({\frac {1}{19}})}{19}}\,}$ ${\displaystyle \infty \,}$
21 Icosidigonal gnomons ${\displaystyle {\frac {x(1+19x)}{(1-x)^{2}}}\,}$ ${\displaystyle 20\,}$ ${\displaystyle 20\,}$ ${\displaystyle P_{22}^{(2)}(m)=\,}$

${\displaystyle m+20{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{20}})-\psi ({\frac {1}{20}})}{20}}\,}$ ${\displaystyle \infty \,}$
22 Icositrigonal gnomons ${\displaystyle {\frac {x(1+20x)}{(1-x)^{2}}}\,}$ ${\displaystyle 21\,}$ ${\displaystyle 21\,}$ ${\displaystyle P_{23}^{(2)}(m)=\,}$

${\displaystyle m+21{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{21}})-\psi ({\frac {1}{21}})}{21}}\,}$ ${\displaystyle \infty \,}$
23 Icositetragonal gnomons ${\displaystyle {\frac {x(1+21x)}{(1-x)^{2}}}\,}$ ${\displaystyle 22\,}$ ${\displaystyle 22\,}$ ${\displaystyle P_{24}^{(2)}(m)=\,}$

${\displaystyle m+22{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{22}})-\psi ({\frac {1}{22}})}{22}}\,}$ ${\displaystyle \infty \,}$
24 Icosipentagonal gnomons ${\displaystyle {\frac {x(1+22x)}{(1-x)^{2}}}\,}$ ${\displaystyle 23\,}$ ${\displaystyle 23\,}$ ${\displaystyle P_{25}^{(2)}(m)=\,}$

${\displaystyle m+23{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{23}})-\psi ({\frac {1}{23}})}{23}}\,}$ ${\displaystyle \infty \,}$
25 Icosihexagonal gnomons ${\displaystyle {\frac {x(1+23x)}{(1-x)^{2}}}\,}$ ${\displaystyle 24\,}$ ${\displaystyle 24\,}$ ${\displaystyle P_{26}^{(2)}(m)=\,}$

${\displaystyle m+24{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{24}})-\psi ({\frac {1}{24}})}{24}}\,}$ ${\displaystyle \infty \,}$
26 Icosiheptagonal gnomons ${\displaystyle {\frac {x(1+24x)}{(1-x)^{2}}}\,}$ ${\displaystyle 25\,}$ ${\displaystyle 25\,}$ ${\displaystyle P_{27}^{(2)}(m)=\,}$

${\displaystyle m+25{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{25}})-\psi ({\frac {1}{25}})}{25}}\,}$ ${\displaystyle \infty \,}$
27 Icosioctagonal gnomons ${\displaystyle {\frac {x(1+25x)}{(1-x)^{2}}}\,}$ ${\displaystyle 26\,}$ ${\displaystyle 26\,}$ ${\displaystyle P_{28}^{(2)}(m)=\,}$

${\displaystyle m+26{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{26}})-\psi ({\frac {1}{26}})}{26}}\,}$ ${\displaystyle \infty \,}$
28 Icosinonagonal gnomons ${\displaystyle {\frac {x(1+26x)}{(1-x)^{2}}}\,}$ ${\displaystyle 27\,}$ ${\displaystyle 27\,}$ ${\displaystyle P_{29}^{(2)}(m)=\,}$

${\displaystyle m+27{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{27}})-\psi ({\frac {1}{27}})}{27}}\,}$ ${\displaystyle \infty \,}$
29 Triacontagonal gnomons ${\displaystyle {\frac {x(1+27x)}{(1-x)^{2}}}\,}$ ${\displaystyle 28\,}$ ${\displaystyle 28\,}$ ${\displaystyle P_{30}^{(2)}(m)=\,}$

${\displaystyle m+28{\binom {m}{2}}\,}$

${\displaystyle {\frac {\psi (m+{\frac {1}{28}})-\psi ({\frac {1}{28}})}{28}}\,}$ ${\displaystyle \infty \,}$

## Table of sequences

Gnomonic numbers sequences
B ${\displaystyle P_{B}^{(1)}(n)\,}$ 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, ...}

## 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 ${\displaystyle \scriptstyle P_{N_{0}}^{(d)}(n)\,}$ 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. 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. 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. 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.