Gnomonic numbers

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

The gnomonic number $\scriptstyle 1+(n-1)b-(1-b)0^{n},\ n\geq 0\,$ is the difference between the n^th regular convex W-gonal number and the (n-1)^th regular convex W-gonal number, where W = b + 2. 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.

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

Formulae

The n^th 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:

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

where

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

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

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

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

where

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

Schläfli-Poincaré (convex) polytope formula

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

For 1-dimensional (d = 1) regular convex polygonal gnomons:

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

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

Recurrence equation

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

with initial conditions

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

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

where

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

Generating function

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

where

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

Order of basis

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

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

where

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

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

numbers of form $\scriptstyle 5n+1\,$ are expressible as 1 term of the form $\scriptstyle 5n+1\,$;

numbers of form $\scriptstyle 5n+2\,$ are expressible as the sum of 2 terms of the form $\scriptstyle 5n+1\,$;

numbers of form $\scriptstyle 5n+3\,$ are expressible as the sum of 3 terms of the form $\scriptstyle 5n+1\,$;

numbers of form $\scriptstyle 5n+4\,$ are expressible as the sum of 4 terms of the form $\scriptstyle 5n+1\,$;

numbers of form $\scriptstyle 5n+0\,$ are expressible as the sum of 5 terms of the form $\scriptstyle 5n+1\,$.

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

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

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

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

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

where

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

Partial sums

$\sum _{n=1}^{m}P_{B}^{(1)}(n)=P_{B+1}^{(2)}(m)=m+(B-1)P_{3}^{(2)}(m-1)=m+(B-1)T_{m-1}=m+(B-1){\binom {m}{2}}$

$=m+(B-1){(m-1)m \over 2}={\frac {m}{2}}[(B-1)m-(B-3)],\,$

where

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

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

Partial sums of reciprocals

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

where

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

and $\scriptstyle \psi (x)\,$ is the digamma function.

Sum of reciprocals

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

where

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

The infinite series diverges logarithmically, i.e.:

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

Table of formulae and values

For $\scriptstyle n\geq 2\,$, we have:

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

where

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

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

Gnomonic numbers formulae and values

B	Name	Formulae $P_{B}^{(1)}(n)=\,$ $\scriptstyle 1+(B-1)(n-1)+(B-2)\ 0^{n},\,$ $\scriptstyle B\geq 2\,$	n = 0	1	2	3	4	5	6	7	8	9	10	11	12	OEIS number
2	Trigonal gnomons Triangular gnomons	$n\,$ Nonnegative integers	0	1	2	3	4	5	6	7	8	9	10	11	12	A001477(n)
3	Tetragonal gnomons Square gnomons	$1+2(n-1)+0^{n}\,$ 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	$1+3(n-1)+2\ 0^{n}\,$	0	1	4	7	10	13	16	19	22	25	28	31	34	A016777(n-1)
5	Hexagonal gnomons	$1+4(n-1)+3\ 0^{n}\,$	0	1	5	9	13	17	21	25	29	33	37	41	45	A016813(n-1)
6	Heptagonal gnomons	$1+5(n-1)+4\ 0^{n}\,$	0	1	6	11	16	21	26	31	36	41	46	51	56	A016861(n-1)
7	Octagonal gnomons	$1+6(n-1)+5\ 0^{n}\,$	0	1	7	13	19	25	31	37	43	49	55	61	67	A016921(n-1)
8	Nonagonal gnomons	$1+7(n-1)+6\ 0^{n}\,$	0	1	8	15	22	29	36	43	50	57	64	71	78	A016993(n-1)
9	Decagonal gnomons	$1+8(n-1)+7\ 0^{n}\,$	0	1	9	17	25	33	41	49	57	65	73	81	89	A017077(n-1)
10	Hendecagonal gnomons	$1+9(n-1)+8\ 0^{n}\,$	0	1	10	19	28	37	46	55	64	73	82	91	100	A017173(n-1)
11	Dodecagonal gnomons	$1+10(n-1)+9\ 0^{n}\,$	0	1	11	21	31	41	51	61	71	81	91	101	111	A017281(n-1)
12	Tridecagonal gnomons	$1+11(n-1)+10\ 0^{n}\,$	0	1	12	23	34	45	56	67	78	89	100	111	122	A017401(n-1)
13	Tetradecagonal gnomons	$1+12(n-1)+11\ 0^{n}\,$	0	1	13	25	37	49	61	73	85	97	109	121	133	A017533(n-1)
14	Pentadecagonal gnomons	$1+13(n-1)+12\ 0^{n}\,$	0	1	14	27	40	53	66	79	92	105	118	131	144	A??????
15	Hexadecagonal gnomons	$1+14(n-1)+13\ 0^{n}\,$	0	1	15	29	43	57	71	85	99	113	127	141	155	A??????
16	Heptadecagonal gnomons	$1+15(n-1)+14\ 0^{n}\,$	0	1	16	31	46	61	76	91	106	121	136	151	166	A??????
17	Octadecagonal gnomons	$1+16(n-1)+15\ 0^{n}\,$	0	1	17	33	49	65	81	97	113	129	145	161	177	A??????
18	Nonadecagonal gnomons	$1+17(n-1)+16\ 0^{n}\,$	0	1	18	35	52	69	86	103	120	137	154	171	188	A??????
19	Icosagonal gnomons	$1+18(n-1)+17\ 0^{n}\,$	0	1	19	37	55	73	91	109	127	145	163	181	199	A??????
20	Icosihenagonal gnomons	$1+19(n-1)+18\ 0^{n}\,$	0	1	20	39	58	77	96	115	134	153	172	191	210	A??????
21	Icosidigonal gnomons	$1+20(n-1)+19\ 0^{n}\,$	0	1	21	41	61	81	101	121	141	161	181	201	221	A??????
22	Icositrigonal gnomons	$1+21(n-1)+20\ 0^{n}\,$	0	1	22	43	64	85	106	127	148	169	190	211	232	A??????
23	Icositetragonal gnomons	$1+22(n-1)+21\ 0^{n}\,$	0	1	23	45	67	89	111	133	155	177	199	221	243	A??????
24	Icosipentagonal gnomons	$1+23(n-1)+22\ 0^{n}\,$	0	1	24	47	70	93	116	139	162	185	208	231	254	A??????
25	Icosihexagonal gnomons	$1+24(n-1)+23\ 0^{n}\,$	0	1	25	49	73	97	121	145	169	193	217	241	265	A??????
26	Icosiheptagonal gnomons	$1+25(n-1)+24\ 0^{n}\,$	0	1	26	51	76	101	126	151	176	201	226	251	276	A??????
27	Icosioctagonal gnomons	$1+26(n-1)+25\ 0^{n}\,$	0	1	27	53	79	105	131	157	183	209	235	261	287	A??????
28	Icosinonagonal gnomons	$1+27(n-1)+26\ 0^{n}\,$	0	1	28	55	82	109	136	163	190	217	244	271	298	A??????
29	Triacontagonal gnomons	$1+28(n-1)+27\ 0^{n}\,$	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 $G_{\{P_{B}^{(1)}(n)\}}(x)=\,$ $\scriptstyle {\frac {x\{1+(B-2)x\}}{(1-x)^{2}}}\,$	Order of basis $g_{\{P_{B}^{(1)}\}}=\,$ $\scriptstyle B-1,\ B\geq 2\,$	Differences $P_{B}^{(1)}(n)-\,$ $P_{B}^{(1)}(n-1)=\,$ $\scriptstyle B-1,\ B\geq 2,\ n\geq 2\,$	Partial sums $\sum _{n=1}^{m}P_{B}^{(1)}(n)=\,$ $P_{B+1}^{(2)}(m)=\,$ $\scriptstyle m+(B-1){\binom {m}{2}},\ B\geq 2\,$	Partial sums of reciprocals $\sum _{n=1}^{m}{1 \over {P_{B}^{(1)}(n)}}=\,$ $\scriptstyle {\frac {\psi (m+{\frac {1}{B-1}})-\psi ({\frac {1}{B-1}})}{B-1}},\ B\geq 2\,$	Sum of Reciprocals $\sum _{n=1}^{\infty }{1 \over {P_{B}^{(1)}(n)}}=$ $\infty \,$
2	Trigonal gnomons Triangular gnomons	${\frac {x}{(1-x)^{2}}}\,$	$1\,$	$1\,$	$P_{3}^{(2)}(m)=\,$ $m+1{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{1}})-\psi ({\frac {1}{1}})}{1}}\,$ $\psi (m+1)+\gamma \,$ $H_{m}\,$	$\infty \,$
3	Tetragonal gnomons Square gnomons	${\frac {x(1+x)}{(1-x)^{2}}}\,$	$2\,$	$2\,$	$P_{4}^{(2)}(m)=\,$ $m+2{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{2}})-\psi ({\frac {1}{2}})}{2}}\,$ ${\frac {\psi (m+{\frac {1}{2}})+\gamma +2\log(2)}{2}}\,$ ${\frac {H_{(m-{\frac {1}{2}})}}{2}}+log(2)\,$	$\infty \,$
4	Pentagonal gnomons	${\frac {x(1+2x)}{(1-x)^{2}}}\,$	$3\,$	$3\,$	$P_{5}^{(2)}(m)=\,$ $m+3{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{3}})-\psi ({\frac {1}{3}})}{3}}\,$	$\infty \,$
5	Hexagonal gnomons	${\frac {x(1+3x)}{(1-x)^{2}}}\,$	$4\,$	$4\,$	$P_{6}^{(2)}(m)=\,$ $m+4{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{4}})-\psi ({\frac {1}{4}})}{4}}\,$	$\infty \,$
6	Heptagonal gnomons	${\frac {x(1+4x)}{(1-x)^{2}}}\,$	$5\,$	$5\,$	$P_{7}^{(2)}(m)=\,$ $m+5{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{5}})-\psi ({\frac {1}{5}})}{5}}\,$	$\infty \,$
7	Octagonal gnomons	${\frac {x(1+5x)}{(1-x)^{2}}}\,$	$6\,$	$6\,$	$P_{8}^{(2)}(m)=\,$ $m+6{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{6}})-\psi ({\frac {1}{6}})}{6}}\,$	$\infty \,$
8	Nonagonal gnomons	${\frac {x(1+6x)}{(1-x)^{2}}}\,$	$7\,$	$7\,$	$P_{9}^{(2)}(m)=\,$ $m+7{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{7}})-\psi ({\frac {1}{7}})}{7}}\,$	$\infty \,$
9	Decagonal gnomons	${\frac {x(1+7x)}{(1-x)^{2}}}\,$	$8\,$	$8\,$	$P_{10}^{(2)}(m)=\,$ $m+8{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{8}})-\psi ({\frac {1}{8}})}{8}}\,$	$\infty \,$
10	Hendecagonal gnomons	${\frac {x(1+8x)}{(1-x)^{2}}}\,$	$9\,$	$9\,$	$P_{11}^{(2)}(m)=\,$ $m+9{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{9}})-\psi ({\frac {1}{9}})}{9}}\,$	$\infty \,$
11	Dodecagonal gnomons	${\frac {x(1+9x)}{(1-x)^{2}}}\,$	$10\,$	$10\,$	$P_{12}^{(2)}(m)=\,$ $m+10{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{10}})-\psi ({\frac {1}{10}})}{10}}\,$	$\infty \,$
12	Tridecagonal gnomons	${\frac {x(1+10x)}{(1-x)^{2}}}\,$	$11\,$	$11\,$	$P_{13}^{(2)}(m)=\,$ $m+11{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{11}})-\psi ({\frac {1}{11}})}{11}}\,$	$\infty \,$
13	Tetradecagonal gnomons	${\frac {x(1+11x)}{(1-x)^{2}}}\,$	$12\,$	$12\,$	$P_{14}^{(2)}(m)=\,$ $m+12{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{12}})-\psi ({\frac {1}{12}})}{12}}\,$	$\infty \,$
14	Pentadecagonal gnomons	${\frac {x(1+12x)}{(1-x)^{2}}}\,$	$13\,$	$13\,$	$P_{15}^{(2)}(m)=\,$ $m+13{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{13}})-\psi ({\frac {1}{13}})}{13}}\,$	$\infty \,$
15	Hexadecagonal gnomons	${\frac {x(1+13x)}{(1-x)^{2}}}\,$	$14\,$	$14\,$	$P_{16}^{(2)}(m)=\,$ $m+14{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{14}})-\psi ({\frac {1}{14}})}{14}}\,$	$\infty \,$
16	Heptadecagonal gnomons	${\frac {x(1+14x)}{(1-x)^{2}}}\,$	$15\,$	$15\,$	$P_{17}^{(2)}(m)=\,$ $m+15{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{15}})-\psi ({\frac {1}{15}})}{15}}\,$	$\infty \,$
17	Octadecagonal gnomons	${\frac {x(1+15x)}{(1-x)^{2}}}\,$	$16\,$	$16\,$	$P_{18}^{(2)}(m)=\,$ $m+16{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{16}})-\psi ({\frac {1}{16}})}{16}}\,$	$\infty \,$
18	Nonadecagonal gnomons	${\frac {x(1+16x)}{(1-x)^{2}}}\,$	$17\,$	$17\,$	$P_{19}^{(2)}(m)=\,$ $m+17{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{17}})-\psi ({\frac {1}{17}})}{17}}\,$	$\infty \,$
19	Icosagonal gnomons	${\frac {x(1+17x)}{(1-x)^{2}}}\,$	$18\,$	$18\,$	$P_{20}^{(2)}(m)=\,$ $m+18{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{18}})-\psi ({\frac {1}{18}})}{18}}\,$	$\infty \,$
20	Icosihenagonal gnomons	${\frac {x(1+18x)}{(1-x)^{2}}}\,$	$19\,$	$19\,$	$P_{21}^{(2)}(m)=\,$ $m+19{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{19}})-\psi ({\frac {1}{19}})}{19}}\,$	$\infty \,$
21	Icosidigonal gnomons	${\frac {x(1+19x)}{(1-x)^{2}}}\,$	$20\,$	$20\,$	$P_{22}^{(2)}(m)=\,$ $m+20{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{20}})-\psi ({\frac {1}{20}})}{20}}\,$	$\infty \,$
22	Icositrigonal gnomons	${\frac {x(1+20x)}{(1-x)^{2}}}\,$	$21\,$	$21\,$	$P_{23}^{(2)}(m)=\,$ $m+21{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{21}})-\psi ({\frac {1}{21}})}{21}}\,$	$\infty \,$
23	Icositetragonal gnomons	${\frac {x(1+21x)}{(1-x)^{2}}}\,$	$22\,$	$22\,$	$P_{24}^{(2)}(m)=\,$ $m+22{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{22}})-\psi ({\frac {1}{22}})}{22}}\,$	$\infty \,$
24	Icosipentagonal gnomons	${\frac {x(1+22x)}{(1-x)^{2}}}\,$	$23\,$	$23\,$	$P_{25}^{(2)}(m)=\,$ $m+23{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{23}})-\psi ({\frac {1}{23}})}{23}}\,$	$\infty \,$
25	Icosihexagonal gnomons	${\frac {x(1+23x)}{(1-x)^{2}}}\,$	$24\,$	$24\,$	$P_{26}^{(2)}(m)=\,$ $m+24{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{24}})-\psi ({\frac {1}{24}})}{24}}\,$	$\infty \,$
26	Icosiheptagonal gnomons	${\frac {x(1+24x)}{(1-x)^{2}}}\,$	$25\,$	$25\,$	$P_{27}^{(2)}(m)=\,$ $m+25{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{25}})-\psi ({\frac {1}{25}})}{25}}\,$	$\infty \,$
27	Icosioctagonal gnomons	${\frac {x(1+25x)}{(1-x)^{2}}}\,$	$26\,$	$26\,$	$P_{28}^{(2)}(m)=\,$ $m+26{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{26}})-\psi ({\frac {1}{26}})}{26}}\,$	$\infty \,$
28	Icosinonagonal gnomons	${\frac {x(1+26x)}{(1-x)^{2}}}\,$	$27\,$	$27\,$	$P_{29}^{(2)}(m)=\,$ $m+27{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{27}})-\psi ({\frac {1}{27}})}{27}}\,$	$\infty \,$
29	Triacontagonal gnomons	${\frac {x(1+27x)}{(1-x)^{2}}}\,$	$28\,$	$28\,$	$P_{30}^{(2)}(m)=\,$ $m+28{\binom {m}{2}}\,$	${\frac {\psi (m+{\frac {1}{28}})-\psi ({\frac {1}{28}})}{28}}\,$	$\infty \,$

Table of sequences

Gnomonic numbers sequences

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

See also

Centered gnomonic numbers

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, 2^nd ed., 1994.