A007678 -id:A007678 - cp's OEIS frontend

A067156 Number of regions in regular n-gon which are 9-gons.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, 123, 0, 0, 88, 45, 0, 0, 0, 0, 0, 51, 0, 0, 0, 165, 0, 114, 0, 118, 120, 61, 124, 0, 192, 195, 66, 67, 272, 138, 0, 568, 360, 146, 222, 600, 0, 231, 156, 237, 800, 567, 410, 664, 84, 255, 344, 174

Offset: 9

Sascha Kurz, Jan 06 2002

nonn

			a(9)=1 because drawing the regular 9-gon with all its diagonals yields 1 9-gon.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 9..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067168, A064869, A067151, A067152, A067153, A067154, A067155, A067157, A067158, A067159.

a(86) and beyond by Scott R. Shannon, Dec 04 2021

A067157 Number of regions in regular n-gon which are 10-gons.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 87, 0, 62, 0, 0, 0, 0, 0, 74, 0, 0, 0, 41, 0, 0, 44, 0, 0, 235, 48, 147, 100, 51, 0, 159, 54, 110, 56, 114, 58, 177, 0, 183, 62, 378, 256, 195, 0, 134, 136, 621, 210, 71, 144, 438, 222, 750, 76, 385, 78, 1185, 80, 648, 82, 830, 336, 935

Offset: 10

Sascha Kurz, Jan 06 2002

nonn

			a(29)=87 because drawing the regular 29-gon with all its diagonals yields 87 10-gons.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 10..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067169, A064869, A067151, A067152, A067153, A067154, A067155, A067156, A067158, A067159.

a(83) and beyond by Scott R. Shannon, Dec 04 2021

A324043 Number of quadrilateral regions into which a figure made up of a row of n adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles.

Original entry on oeis.org

0, 2, 14, 34, 90, 154, 288, 462, 742, 1038, 1512, 2074, 2904, 3774, 4892, 6154, 7864, 9662, 12022, 14638, 17786, 20998, 25024, 29402, 34672, 40038, 46310, 53038, 61090, 69454, 79344, 89890, 101792, 113854, 127476, 141866, 158428, 175182, 193760, 213274, 235444, 258182, 283858, 310750, 339986

Offset: 1

Jinyuan Wang, May 01 2019

nonn

A row of n adjacent congruent rectangles can only be divided into triangles (cf. A324042) or quadrilaterals when drawing diagonals. Proof is given in Alekseyev et al. (2015) under the transformation described in A306302.

			For k adjacent congruent rectangles, the number of quadrilateral regions in the j-th rectangle is:
k\j|  1   2   3   4   5   6   7  ...
---+--------------------------------
1  |  0,  0,  0,  0,  0,  0,  0, ...
2  |  1,  1,  0,  0,  0,  0,  0, ...
3  |  3,  8,  3,  0,  0,  0,  0, ...
4  |  5, 12, 12,  5,  0,  0,  0, ...
5  |  7, 22, 32, 22,  7,  0,  0, ...
6  |  9, 28, 40, 40, 28,  9,  0, ...
7  | 11, 38, 58, 74, 58, 38, 11, ...
...
a(4) = 5 + 12 + 12 + 5 = 34.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh, On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165.
Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.
Robert Israel, Maple program.
Jinyuan Wang, Illustration for n = 1, 2, 3, 4, 5.

Cf. A007678, A108914, A114043, A115005, A177719, A306302, A324042, A333286, A333287, A333288.

See Robert Israel link.
There are also Maple programs for both A306302 and A324042. Then a := n -> A306302(n) - A324042(n); # N. J. A. Sloane, Mar 04 2020

Table[Sum[Sum[(Boole[GCD[i, j] == 1] - 2 * Boole[GCD[i, j] == 2]) * (n + 1 - i) * (n + 1 - j), {j, 1, n}], {i, 1, n}] - n^2, {n, 1, 45}] (* Joshua Oliver, Feb 05 2020 *)

{ A324043(n) = sum(i=1, n, sum(j=1, n, ( (gcd(i, j)==1) - 2*(gcd(i,j)==2) ) * (n+1-i) * (n+1-j) )) - n^2; } \\ Max Alekseyev, Jul 08 2019

from sympy import totient
def A324043(n): return 0 if n==1 else -2*(n-1)**2 + sum(totient(i)*(n+1-i)*(7*i-2*n-2) for i in range(2,n//2+1)) + sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(n//2+1,n+1)) # Chai Wah Wu, Aug 16 2021

a(n) = A115005(n+1) - A177719(n+1) - n - 1 = Sum_{i,j=1..n; gcd(i,j)=1} (n+1-i)*(n+1-j) - 2*Sum_{i,j=1..n; gcd(i,j)=2} (n+1-i)*(n+1-j) - n^2. - Max Alekseyev, Jul 08 2019
a(n) = A306302(n) - A324042(n).
For n>1, a(n) = -2(n-1)^2 + Sum_{i=2..floor(n/2)} (n+1-i)*(7i-2n-2)*phi(i) + Sum_{i=floor(n/2)+1..n} (n+1-i)*(2n+2-i)*phi(i). - Chai Wah Wu, Aug 16 2021

a(8)-a(23) from Robert Israel, Jul 07 2019

Terms a(24) onward from Max Alekseyev, Jul 08 2019

A331906 The number of regions inside a pentagram formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

40, 1100, 7330, 25540, 65930, 136200, 263010, 458410, 740550, 1142740, 1681640, 2400970, 3338850, 4495510, 5962220, 7736150, 9924580, 12442880, 15527670, 19132140, 23301600, 28070620, 33585800, 39919140, 47157510, 55209750, 64185300, 74311940, 85731780, 98167130

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 31 2020

nonn

The terms are from numeric computation - no formula for a(n) is currently known.

Scott R. Shannon, Pentagram regions for n = 1.
Scott R. Shannon, Pentagram regions for n = 2.
Scott R. Shannon, Pentagram regions for n = 3.
Scott R. Shannon, Pentagram regions for n = 4.
Scott R. Shannon, Pentagram regions for n = 5.
Scott R. Shannon, Pentagram regions for n = 6.
Scott R. Shannon, Pentagram regions with random distance-based coloring for n = 1.
Scott R. Shannon, Pentagram regions with random distance-based coloring for n = 2.
Scott R. Shannon, Pentagram regions with random distance-based coloring for n = 3.
Scott R. Shannon, Pentagram regions with random distance-based coloring for n = 4.
Scott R. Shannon, Pentagram regions with random distance-based coloring for n = 5.
Eric Weisstein's World of Mathematics, Pentagram.

Cf. A331907 (n-gons), A333117 (vertices), A333118 (edges), A007678, A092867, A331452.

a(7)-a(30) from Lars Blomberg, May 06 2020

A342222 a(n) is the smallest m such that a regular m-gon with all diagonals drawn contains a cell with n sides, or a(n) = -1 if no such m exists.

Original entry on oeis.org

3, 6, 5, 9, 7, 13, 9, 29, 11, 40, 13, 43, 15, 212, 17, 231, 19

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Mar 06 2021

Theorem: If n is odd then a(n) = n.
Proof. (i) If n is odd then the central cell in a regular n-gon with all diagonals drawn is a smaller regular n-gon. So if n is odd, then a(n) <= n.
(ii) Suppose a convex m-gon, not necessarily regular, with all diagonals drawn has a cell with e edges. Each edge when extended meets two vertices, so at most 2e vertices are involved in defining the boundary of that cell.
On the other hand no vertex can define more than two edges of the cell, so 2e <= 2m, so e <= m. So to get an n-sided cell, we need at least n vertices. So a(n) >= n. QED.
If a(20) > 0 it is greater than 765 - Scott R. Shannon, Nov 30 2021

			Examining the images in A007678, for example Michael Rubinstein's illustration, or the images shown here, we see that the first occurrence of a five-sided cell is for m = 5, so a(5) = 5. The first time we see a four-sided cell is for m = 6, so a(4) = 6.

Martin Balko, Anna Brötzner, Fabian Klute, and Josef Tkadlec, Faces in Rectilinear Drawings of Complete Graphs, 40th European Workshop on Computational Geometry, Ioannina, Greece, March 13-15, 2024. See pp. 4, 7.
Scott R. Shannon, Image for a(3) = 3.
Scott R. Shannon, Image for a(4) = 6.
Scott R. Shannon, Image for a(5) = 5.
Scott R. Shannon, Image for a(6) = a(9) = 9.
Scott R. Shannon, Image for a(7) = 7.
Scott R. Shannon, Image for a(8) = a(13) = 13.
Scott R. Shannon, Image for a(10) = 29.
Scott R. Shannon, Image for a(11) = 11.
Scott R. Shannon, Image for a(12) = 40.
Scott R. Shannon, Image for a(14) = 43.
Scott R. Shannon, Image for a(15) = 15.

Cf. A007678, A331450, A331451, A007569, A135565.

See also A341729 and A341730 for the maximum number of sides in any cell.

a(16)-a(19) added by Scott R. Shannon, Mar 14 2021

A067158 Number of regions in regular n-gon which are 11-gons.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 29, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 71, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 99, 0, 0, 102, 103, 0, 0, 0, 0, 108, 0

Offset: 11

Sascha Kurz, Jan 06 2002

nonn

			a(11)=1 because drawing the regular 11-gon with all its diagonals yields 1 11-gon.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 11..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A064869, A067151, A067152, A067153, A067154, A067155, A067156, A067157, A067159.

a(110) and beyond by Scott R. Shannon, Dec 04 2021

A067159 Number of regions in regular n-gon which are 12-gons.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 40, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 51, 0, 53, 0, 0, 0, 0, 0, 59, 0, 61, 0, 63, 0, 65, 0, 0, 0, 0, 0, 0, 0, 73, 0, 225, 76, 0, 0, 0, 0, 0, 0, 0, 0, 170, 0, 0, 0, 0, 0, 273, 0, 93, 0, 95, 0, 97, 0, 0, 0, 101, 102, 412

Offset: 12

Sascha Kurz, Jan 06 2002

nonn

			a(40)=40 because drawing the regular 40-gon with all its diagonals yields 40 12-gons.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 12..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A064869, A067151, A067152, A067153, A067154, A067155, A067156, A067157, A067158.

a(104) and beyond by Scott R. Shannon, Dec 04 2021

A331448 a(0) = 0 by convention; for n>0, a(n) is the number of edges in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

Original entry on oeis.org

0, 8, 92, 596, 1936, 6020, 11088, 26260, 42144, 72296, 107832, 183340, 222940, 371100, 466936, 609916, 804504, 1139632, 1288536, 1813288, 2012676, 2536572, 3142008, 3997580, 4230340, 5430444, 6331892, 7360512, 8262568, 10367804

Offset: 0

N. J. A. Sloane, Jan 23 2020, based on a comment from Michael De Vlieger in A255011

nonn

By Euler's formula, a(n) = A331449(n) + A255011(n) - 1.

Zhao Hui Du, Table of n, a(n) for n = 0..136 (terms 0..52 from Lars Blomberg).

Cf. A255011, A331449.

For the circular analog see A006533, A007678, A007569, A135565.

a(11)-a(29) from Giovanni Resta, Jan 28 2020

A331456 Number of regions in an equal-armed cross with arms of length n (see Comments for definition).

Original entry on oeis.org

4, 104, 568, 1900, 4808, 10180, 19180, 33132, 53628, 82432, 121448, 172948, 239356, 323168, 427272, 554892, 709476, 893772, 1111588, 1367292, 1664604, 2008240, 2402560, 2852532, 3363280, 3938712, 4585568, 5308720, 6112736, 7006068, 7994412, 9084788, 10281812

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Jan 28 2020

nonn

This cross of height n consists of a central square with 4 arms of length n.
There are 4n+1 squares in all. The number of vertices is 8n+4.
Now join every pair of vertices by a line segment, provided the line does not extend beyond the boundary of the cross. The sequence gives the number of regions in the resulting figure.
See A337641 for information about these regions, their numbers of sides, their coordinates, and for further illustrations. - N. J. A. Sloane, Sep 17 2020

Lars Blomberg, Table of n, a(n) for n = 0..48
Scott R. Shannon, Colored illustration for a(0).
Scott R. Shannon, Colored illustration for a(1).
Scott R. Shannon, Colored illustration for a(2).
Scott R. Shannon, Colored illustration for a(3).
Scott R. Shannon, Colored illustration for a(4).
Scott R. Shannon, Colored illustration for a(5).
Scott R. Shannon, Colored illustration for a(9).
Scott R. Shannon, Colored illustration for a(1) classifying nodes and cells.
Scott R. Shannon, Colored illustration for a(2) classifying nodes and cells.
Scott R. Shannon, Colored illustration for a(3) classifying nodes and cells.
Scott R. Shannon, Colored illustration for a(4) classifying nodes and cells.
Scott R. Shannon, Colored illustration for a(5) classifying nodes and cells.
Scott R. Shannon, Colored illustration for a(6) classifying nodes and cells.
N. J. A. Sloane, Illustration for a(1). (One of the "arms" has been cropped by the scanner, but all four arms are the same.)
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.

Cf. A333035 (vertices), A333036 (edges), A333037 (n-gons), A337641.
See A331455 for a different family of crosses.
A331452 is a similar sequence for a rectangular region; A007678 for a polygonal region.
Cf. A331458.

a(11) and beyond from Lars Blomberg, May 30 2020

A333075 The number of regions inside an octagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

80, 1488, 9312, 31552, 83432, 174816, 339816, 584176, 953416, 1463936, 2173976, 3074784, 4294080, 5790816, 7664880, 9952944, 12757088, 16036096, 20013696, 24577760, 29973528, 36161472, 43314312, 51334672

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 07 2020

The terms are from numeric computation - no formula for a(n) is currently known.

Scott R. Shannon, Octagon regions for n = 1.
Scott R. Shannon, Octagon regions for n = 2.
Scott R. Shannon, Octagon regions for n = 3.
Scott R. Shannon, Octagon regions for n = 4.
Scott R. Shannon, Octagon regions using random distance-based coloring for n = 2.
Scott R. Shannon, Octagon regions using random distance-based coloring for n = 3.
Scott R. Shannon, Octagon regions using random distance-based coloring for n = 4.
Wikipedia, Octagon.

Cf. A333076 (n-gons), A333109 (vertices), A333110 (edges), A007678, A092867, A331452, A331931.

a(7)-a(24) from Lars Blomberg, May 14 2020

cp's OEIS Frontend

A067156 Number of regions in regular n-gon which are 9-gons.

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A067157 Number of regions in regular n-gon which are 10-gons.

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A324043 Number of quadrilateral regions into which a figure made up of a row of n adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles.

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A331906 The number of regions inside a pentagram formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

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A342222 a(n) is the smallest m such that a regular m-gon with all diagonals drawn contains a cell with n sides, or a(n) = -1 if no such m exists.

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A067158 Number of regions in regular n-gon which are 11-gons.

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A067159 Number of regions in regular n-gon which are 12-gons.

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A331448 a(0) = 0 by convention; for n>0, a(n) is the number of edges in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

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A331456 Number of regions in an equal-armed cross with arms of length n (see Comments for definition).

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