Centered gnomonic numbers

The centered gnomonic numbers are the centered equidistributions of $\scriptstyle a+nb,\ (a,b)=1,\ n\geq 0\,$ 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:

$1,\ n=0,\,$

$(b+2)+2(n-1)b=b+2[1+(n-1)b],\ n\geq 1\,$, when b is odd positive integer,

$(b+1)+(n-1)b=b+[1+(n-1)b],\ n\geq 1\,$, when b is even positive integer;

or more compactly:

$1,\ n=0,\,$

$b+{\bigg (}{\frac {3-(-1)^{b}}{2}}{\bigg )}[1+(n-1)b],\ n\geq 1;\,$

or even more compactly:

$0^{n}+(1-0^{n}){\bigg \{}b+{\bigg (}{\frac {3-(-1)^{b}}{2}}{\bigg )}[1+(n-1)b]{\bigg \}},\ n\geq 0.\,$

The two cases are explained as follow:

• when b is even positive integer, $\scriptstyle 1+nb,\ n\geq 0\,$ is always odd, so when centered, any previous dot is masked by a new dot, thus we get $\scriptstyle 1+nb,\ n\geq 0\,$ dots as a result;

• when b is odd positive integer, $\scriptstyle 1+nb,\ n\geq 0\,$ 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 $\scriptstyle 1,\ n=0;\ (b+2)+2(n-1)b,\ n\geq 1\,$ dots as a result.

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

Formulae

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

$\,_{c}P_{B}^{(1)}(n)=0^{n}+(1-0^{n}){\bigg \{}B-1+{\bigg (}{\frac {3+(-1)^{B}}{2}}{\bigg )}[1+(B-1)(n-1)]{\bigg \}},\ n\geq 0.\,$

where

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

The choice of $\scriptstyle B=b+1\geq 2\,$ 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.

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

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

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

Recurrence equation

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

with initial conditions

$\,_{c}P_{B}^{(1)}(1)=B+{\bigg (}{\frac {1+(-1)^{B}}{2}}{\bigg )}\,$

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

where

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

Generating function

$G_{\{\,_{c}P_{B}^{(1)}(n)\}}(x)={\frac {1+{\bigg (}B-{\bigg (}{\frac {3-(-1)^{B}}{2}}{\bigg )}{\bigg )}x+{\bigg (}{\frac {1+(-1)^{B}}{2}}{\bigg )}(B-2)x^{2}}{(1-x)^{2}}},\,$

where

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

Order of basis

The order of basis of centered gnomonic numbers is:

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

where

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

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

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

$\,_{c}P_{B}^{(1)}(n)-\,_{c}P_{B}^{(1)}(n-1)=B-1+{\bigg (}{\frac {1+(-1)^{B}}{2}}{\bigg )},\ n=1,\,$

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

where

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

Partial sums

$\sum _{n=0}^{m}\,_{c}P_{B}^{(1)}(n)=$

where

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

Partial sums of reciprocals

$\sum _{n=0}^{m}{\frac {1}{\,_{c}P_{B}^{(1)}(n)}}=,\,$

where

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

Sum of reciprocals

$\sum _{n=0}^{\infty }{\frac {1}{\,_{c}P_{B}^{(1)}(n)}}=\infty ,\,$

where

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

The infinite series diverges logarithmically, i.e.:

$\sum _{n=0}^{m}{\frac {1}{\,_{c}P_{B}^{(1)}(n)}}\sim O(log(m)),\,$ as $m\to \infty .\,$

Table of formulae and values

Centered gnomonic numbers formulae and values

B	Name	Formulae $\,_{c}P_{B}^{(1)}(n)=\,$ $\scriptstyle 0^{n}+(1-0^{n}){\big \{}B-1+{\big (}{\frac {3+(-1)^{B}}{2}}{\big )}[1+(B-1)(n-1)]{\big \}},\,$ $\scriptstyle \ B\geq 2,\ n\geq 0\,$	n = 0	1	2	3	4	5	6	7	8	9	10	11	12	OEIS number
2		$0^{n}+(1-0^{n}){\big \{}1+2[1+(1)(n-1)]{\big \}}\,$ Odd numbers	1	3	5	7	9	11	13	15	17	19	21	23	25	A005408(n)
3		$0^{n}+(1-0^{n}){\big \{}2+[1+(2)(n-1)]{\big \}}\,$ Odd numbers	1	3	5	7	9	11	13	15	17	19	21	23	25	A005408(n)
4		$0^{n}+(1-0^{n}){\big \{}3+2[1+(3)(n-1)]{\big \}}\,$	1	5	11	17	23	29	35	41	47	53	59	65	71	A101328(n+2)
5		$0^{n}+(1-0^{n}){\big \{}4+[1+(4)(n-1)]{\big \}}\,$	1	5	9	13	17	21	25	29	33	37	41	45	49	A016813(n)
6		$0^{n}+(1-0^{n}){\big \{}5+2[1+(5)(n-1)]{\big \}}\,$	1	7	17	27	37	47	57	67	77	87	97	107	117	A160912(n+1)
7		$0^{n}+(1-0^{n}){\big \{}6+[1+(6)(n-1)]{\big \}}\,$	1	7	13	19	25	31	37	43	49	55	61	67	73	A016921(n)
8		$0^{n}+(1-0^{n}){\big \{}7+2[1+(7)(n-1)]{\big \}}\,$	1	9	23	37	51	65	79	93	107	121	135	149	163	Axxxxxx
9		$0^{n}+(1-0^{n}){\big \{}8+[1+(8)(n-1)]{\big \}}\,$	1	9	17	25	33	41	49	57	65	73	81	89	97	A017077(n)
10		$0^{n}+(1-0^{n}){\big \{}9+2[1+(9)(n-1)]{\big \}}\,$	1	11	29	47	65	83	101	119	137	155	173	191	209	Axxxxxx
11		$0^{n}+(1-0^{n}){\big \{}10+[1+(10)(n-1)]{\big \}}\,$	1	11	21	31	41	51	61	71	81	91	101	111	121	A017281(n)
12		$0^{n}+(1-0^{n}){\big \{}11+2[1+(11)(n-1)]{\big \}}\,$	1	13	35	57	79	101	123	145	167	189	211	233	255	Axxxxxx
13		$0^{n}+(1-0^{n}){\big \{}12+[1+(12)(n-1)]{\big \}}\,$	1	13	25	37	49	61	73	85	97	109	121	133	145	A017533(n)
14		$0^{n}+(1-0^{n}){\big \{}13+2[1+(13)(n-1)]{\big \}}\,$	1	15	41	67	93	119	145	171	197	223	249	275	301	Axxxxxx
15		$0^{n}+(1-0^{n}){\big \{}14+[1+(14)(n-1)]{\big \}}\,$	1	15	29	43	57	71	85	99	113	127	141	155	169	Axxxxxx
16		$0^{n}+(1-0^{n}){\big \{}15+2[1+(15)(n-1)]{\big \}}\,$	1	17	47	77	107	137	167	197	227	257	287	317	347	Axxxxxx
17		$0^{n}+(1-0^{n}){\big \{}16+[1+(16)(n-1)]{\big \}}\,$	1	17	33	49	65	81	97	113	129	145	161	177	193	Axxxxxx
18		$0^{n}+(1-0^{n}){\big \{}17+2[1+(17)(n-1)]{\big \}}\,$	1	19	53	87	121	155	189	223	257	291	325	359	393	Axxxxxx
19		$0^{n}+(1-0^{n}){\big \{}18+[1+(18)(n-1)]{\big \}}\,$	1	19	37	55	73	91	109	127	145	163	181	199	217	Axxxxxx
20		$0^{n}+(1-0^{n}){\big \{}19+2[1+(19)(n-1)]{\big \}}\,$	1	21	59	97	135	173	211	249	287	325	363	401	439	Axxxxxx
21		$0^{n}+(1-0^{n}){\big \{}20+[1+(20)(n-1)]{\big \}}\,$	1	21	41	61	81	101	121	141	161	181	201	221	241	Axxxxxx
22		$0^{n}+(1-0^{n}){\big \{}21+2[1+(21)(n-1)]{\big \}}\,$	1	23	65	107	149	191	233	275	317	359	401	443	485	Axxxxxx
23		$0^{n}+(1-0^{n}){\big \{}22+[1+(22)(n-1)]{\big \}}\,$	1	23	45	67	89	111	133	155	177	199	221	243	265	Axxxxxx
24		$0^{n}+(1-0^{n}){\big \{}23+2[1+(23)(n-1)]{\big \}}\,$	1	25	71	117	163	209	255	301	347	393	439	485	531	Axxxxxx
25		$0^{n}+(1-0^{n}){\big \{}24+[1+(24)(n-1)]{\big \}}\,$	1	25	49	73	97	121	145	169	193	217	241	265	289	Axxxxxx
26		$0^{n}+(1-0^{n}){\big \{}25+2[1+(25)(n-1)]{\big \}}\,$	1	27	77	127	177	227	277	327	377	427	477	527	577	Axxxxxx
27		$0^{n}+(1-0^{n}){\big \{}26+[1+(26)(n-1)]{\big \}}\,$	1	27	53	79	105	131	157	183	209	235	261	287	313	Axxxxxx
28		$0^{n}+(1-0^{n}){\big \{}27+2[1+(27)(n-1)]{\big \}}\,$	1	29	83	137	191	245	299	353	407	461	515	569	623	Axxxxxx
29		$0^{n}+(1-0^{n}){\big \{}28+[1+(28)(n-1)]{\big \}}\,$	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 $G_{\{\,_{c}P_{B}^{(1)}(n)\}}(x)=\,$ $\scriptstyle {\frac {1+{\big (}B-{\big (}{\frac {3-(-1)^{B}}{2}}{\big )}{\big )}x+{\big (}{\frac {1+(-1)^{B}}{2}}{\big )}(B-2)x^{2}}{(1-x)^{2}}}\,$	Order of basis $g_{\{\,_{c}P_{B}^{(1)}\}}=\,$	Differences $\,_{c}P_{B}^{(1)}(n)-\,$ $\,_{c}P_{B}^{(1)}(n-1)=\,$	Partial sums $\sum _{n=0}^{m}\,_{c}P_{B}^{(1)}(n)=\,$	Partial sums of reciprocals $\sum _{n=0}^{m}{1 \over {\,_{c}P_{B}^{(1)}(n)}}=\,$	Sums of reciprocals $\sum _{n=0}^{\infty }{1 \over {\,_{c}P_{B}^{(1)}(n)}}=$ $\infty \,$
2		${\frac {1+x}{(1-x)^{2}}}\,$	$2\,$	$2\,$	$P_{4}^{(2)}(m+1)=\,$ $m+1+2{\binom {m+1}{2}}\,$	${\frac {\psi (m+1+{\frac {1}{2}})-\psi ({\frac {1}{2}})}{2}}\,$ ${\frac {\psi (m+1+{\frac {1}{2}})+\gamma +2\log(2)}{2}}\,$ ${\frac {H_{(m+1-{\frac {1}{2}})}}{2}}+log(2)\,$	$\infty \,$
3		${\frac {1+x}{(1-x)^{2}}}\,$	$2\,$	$2\,$	$P_{4}^{(2)}(m+1)=\,$ $m+1+2{\binom {m+1}{2}}\,$	${\frac {\psi (m+1+{\frac {1}{2}})-\psi ({\frac {1}{2}})}{2}}\,$ ${\frac {\psi (m+1+{\frac {1}{2}})+\gamma +2\log(2)}{2}}\,$ ${\frac {H_{(m+1-{\frac {1}{2}})}}{2}}+log(2)\,$	$\infty \,$
4		${\frac {1+3x+2x^{2}}{(1-x)^{2}}}\,$	$\,$	$\,$	$\,$	$\,$	$\infty \,$
5		${\frac {1+3x}{(1-x)^{2}}}\,$	$4\,$	$4\,$	$P_{6}^{(2)}(m+1)=\,$ $m+1+4{\binom {m+1}{2}}\,$	${\frac {\psi (m+1+{\frac {1}{4}})-\psi ({\frac {1}{4}})}{4}}\,$	$\infty \,$
6		${\frac {1+5x+4x^{2}}{(1-x)^{2}}}\,$	$\,$	$\,$	$\,$	$\,$	$\infty \,$
7		${\frac {1+5x}{(1-x)^{2}}}\,$	$6\,$	$6\,$	$P_{8}^{(2)}(m+1)=\,$ $m+1+6{\binom {m+1}{2}}\,$	${\frac {\psi (m+1+{\frac {1}{6}})-\psi ({\frac {1}{6}})}{6}}\,$	$\infty \,$
8		${\frac {1+7x+6x^{2}}{(1-x)^{2}}}\,$	$\,$	$\,$	$\,$	$\,$	$\infty \,$
9		${\frac {1+7x}{(1-x)^{2}}}\,$	$8\,$	$8\,$	$P_{10}^{(2)}(m+1)=\,$ $m+1+8{\binom {m+1}{2}}\,$	${\frac {\psi (m+1+{\frac {1}{8}})-\psi ({\frac {1}{8}})}{8}}\,$	$\infty \,$
10		${\frac {1+9x+8x^{2}}{(1-x)^{2}}}\,$	$\,$	$\,$	$\,$	$\,$	$\infty \,$
11		${\frac {1+9x}{(1-x)^{2}}}\,$	$10\,$	$10\,$	$P_{12}^{(2)}(m+1)=\,$ $m+1+10{\binom {m+1}{2}}\,$	${\frac {\psi (m+1+{\frac {1}{10}})-\psi ({\frac {1}{10}})}{10}}\,$	$\infty \,$
12		${\frac {1+11x+10x^{2}}{(1-x)^{2}}}\,$	$\,$	$\,$	$\,$	$\,$	$\infty \,$
13		${\frac {1+11x}{(1-x)^{2}}}\,$	$12\,$	$12\,$	$P_{14}^{(2)}(m+1)=\,$ $m+1+12{\binom {m+1}{2}}\,$	${\frac {\psi (m+1+{\frac {1}{12}})-\psi ({\frac {1}{12}})}{12}}\,$	$\infty \,$
14		${\frac {1+13x+12x^{2}}{(1-x)^{2}}}\,$	$\,$	$\,$	$\,$	$\,$	$\infty \,$
15		${\frac {1+13x}{(1-x)^{2}}}\,$	$14\,$	$14\,$	$P_{16}^{(2)}(m+1)=\,$ $m+1+14{\binom {m+1}{2}}\,$	${\frac {\psi (m+1+{\frac {1}{14}})-\psi ({\frac {1}{14}})}{14}}\,$	$\infty \,$
16		${\frac {1+15x+14x^{2}}{(1-x)^{2}}}\,$	$\,$	$\,$	$\,$	$\,$	$\infty \,$
17		${\frac {1+15x}{(1-x)^{2}}}\,$	$16\,$	$16\,$	$P_{18}^{(2)}(m+1)=\,$ $m+1+16{\binom {m+1}{2}}\,$	${\frac {\psi (m+1+{\frac {1}{16}})-\psi ({\frac {1}{16}})}{16}}\,$	$\infty \,$
18		${\frac {1+17x+16x^{2}}{(1-x)^{2}}}\,$	$\,$	$\,$	$\,$	$\,$	$\infty \,$
19		${\frac {1+17x}{(1-x)^{2}}}\,$	$18\,$	$18\,$	$P_{20}^{(2)}(m+1)=\,$ $m+1+18{\binom {m+1}{2}}\,$	${\frac {\psi (m+1+{\frac {1}{18}})-\psi ({\frac {1}{18}})}{18}}\,$	$\infty \,$
20		${\frac {1+19x+18x^{2}}{(1-x)^{2}}}\,$	$\,$	$\,$	$\,$	$\,$	$\infty \,$
21		${\frac {1+19x}{(1-x)^{2}}}\,$	$20\,$	$20\,$	$P_{22}^{(2)}(m+1)=\,$ $m+1+20{\binom {m+1}{2}}\,$	${\frac {\psi (m+1+{\frac {1}{20}})-\psi ({\frac {1}{20}})}{20}}\,$	$\infty \,$
22		${\frac {1+21x+20x^{2}}{(1-x)^{2}}}\,$	$\,$	$\,$	$\,$	$\,$	$\infty \,$
23		${\frac {1+21x}{(1-x)^{2}}}\,$	$22\,$	$22\,$	$P_{24}^{(2)}(m+1)=\,$ $m+1+22{\binom {m+1}{2}}\,$	${\frac {\psi (m+1+{\frac {1}{22}})-\psi ({\frac {1}{22}})}{22}}\,$	$\infty \,$
24		${\frac {1+23x+22x^{2}}{(1-x)^{2}}}\,$	$\,$	$\,$	$\,$	$\,$	$\infty \,$
25		${\frac {1+23x}{(1-x)^{2}}}\,$	$24\,$	$24\,$	$P_{26}^{(2)}(m+1)=\,$ $m+1+24{\binom {m+1}{2}}\,$	${\frac {\psi (m+1+{\frac {1}{24}})-\psi ({\frac {1}{24}})}{24}}\,$	$\infty \,$
26		${\frac {1+25x+24x^{2}}{(1-x)^{2}}}\,$	$\,$	$\,$	$\,$	$\,$	$\infty \,$
27		${\frac {1+25x}{(1-x)^{2}}}\,$	$26\,$	$26\,$	$P_{28}^{(2)}(m+1)=\,$ $m+1+26{\binom {m+1}{2}}\,$	${\frac {\psi (m+1+{\frac {1}{26}})-\psi ({\frac {1}{26}})}{26}}\,$	$\infty \,$
28		${\frac {1+27x+26x^{2}}{(1-x)^{2}}}\,$	$\,$	$\,$	$\,$	$\,$	$\infty \,$
29		${\frac {1+27x}{(1-x)^{2}}}\,$	$28\,$	$28\,$	$P_{30}^{(2)}(m+1)=\,$ $m+1+28{\binom {m+1}{2}}\,$	${\frac {\psi (m+1+{\frac {1}{28}})-\psi ({\frac {1}{28}})}{28}}\,$	$\infty \,$

Table of sequences

Centered gnomonic numbers sequences

B	$\,_{c}P_{B}^{(1)}(n),\ n\geq 0\,$ 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

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.