A007678 -id:A007678 - cp's OEIS frontend

A341877 The number of regions inside a vesica piscis formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

1, 4, 24, 84, 236, 508, 1048, 1820, 3110, 4644, 7428, 10520, 15108, 20288, 27576, 35632, 46652, 58548, 74156, 89992, 112288, 135064, 163684, 193572, 230884, 269612, 316846, 365496, 425060, 485392, 558744, 632888, 721636, 811724, 918040, 1025224, 1151904, 1279188, 1427720, 1575496

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 22 2021

nonn

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

Scott R. Shannon, Regions for n=4.
Scott R. Shannon, Regions for n=5.
Scott R. Shannon, Regions for n=10.
Scott R. Shannon, Regions for n=15.
Scott R. Shannon, Regions for n=16.
Wikipedia, Vesica piscis.

Cf. A341878 (vertices), A342152 (edges), A342153 (n-gons), A007678, A092867, A340639, A340685.

A344907 Number of polygon edges formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line.

Original entry on oeis.org

0, 3, 30, 189, 684, 1815, 3978, 7665, 13464, 22059, 34230, 50853, 72900, 101439, 137634, 182745, 238128, 305235, 385614, 480909, 592860, 723303, 874170, 1047489, 1245384, 1470075, 1723878, 2009205, 2328564, 2684559, 3079890, 3517353, 3999840, 4530339, 5111934, 5747805, 6441228, 7195575

Offset: 1

Scott R. Shannon, Jun 02 2021

This sequences gives the number of polygon edges formed when connecting every pair of vertices of a regular polygon, with an odd number of vertices, by an infinite line.
A bisection of A344899. - N. J. A. Sloane, Sep 12 2021
See A344857 for other examples and images of the polygons.

			a(3) = 30 as the five connected vertices form a pentagon with fives lines along the pentagon's edges, fifteen lines inside forming eleven polygons, and ten lines outside forming another five triangles. In all these sixteen polygons form thirty edges. Twenty infinite edges between the outer unbounded regions are also formed.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A344899 (number of edges for all n-gons), A344866 (number of polygon), A146212, A344857, A344311, A007678, A331450, A344938.

A345649 Number of vertices formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

Original entry on oeis.org

0, 5, 61, 465, 1585, 5257, 9749, 24025, 38381, 67177, 100889, 176005, 210033, 360877, 450349, 589581, 779541, 1127509, 1251805, 1806061, 1970129, 2504401, 3116945, 4017701, 4163753, 5433657, 6335589

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Jun 21 2021

See A345459 for images of the polygons.

Cf. A345459 (number of polygons), A345650 (number of edges), A255011, A345648, A344993, A344857, A092098, A007678.

a(n) = A345650(n) - A345459(n) + 1.

A345650 Number of polygon edges formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

Original entry on oeis.org

0, 8, 140, 1032, 3608, 11308, 21892, 51000, 83404, 143900, 217728, 367132, 456008, 749328, 952236, 1251056, 1649708, 2327232, 2653900, 3717444, 4158448, 5243680, 6488208, 8241988, 8780976, 11235028, 13116156

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Jun 21 2021

See A345459 for images of the polygons.

Cf. A331448 (number inside the square), A345459 (number of polygons), A345649 (number of vertices), A255011, A345648, A344993, A344857, A092098, A007678.

a(n) = A345459(n) + A345649(n) - 1.

A359116 Mark the points of the Farey series F_n on a strip of paper and wrap it around a circle of circumference 1 so the endpoints 0 and 1 coincide; draw a chord between every pair of the Farey points; a(n) is the number of vertices in the resulting graph.

Original entry on oeis.org

1, 2, 5, 19, 208, 480, 3011, 7185, 20169, 35438, 111232, 162062, 422841, 633226, 1024370, 1576122, 3315790, 4240974, 8204951, 10654475, 15310713

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Dec 16 2022

Let F_n denote the Farey series of order n: F_1 = [0, 1]; F_2 = [0, 1/2, 1]; F_3 = [0, 1/3, 1/2, 2/3, 1], F_4 = [0, 1/4, 1/3, 1/2, 2/3, 3/4, 1], etc. In general F_n consists of the points i/j with 1 <= j <= n, 0 <= i <= j, gcd(i,j) = 1, sorted and duplicates removed. Alternatively, F_n = [A006842(n,k)/A006843(n,k), k = 1..A005728(n)].
The number of terms in F_n is A005728(n). Since the endpoints coincide when we wrap the series around the circle, there are M = A005728(n) - 1 vertices on the circumference.
The planar graph we are studying, denoted by FR(n), is formed by drawing a chord between every pair of the M boundary points.  FR stands for Farey Ring, a term suggested by the fairy rings found in nature.
FR(n) is analogous to the planar graph formed by drawing chords between every pair of vertices of a regular n-gon, and studied in A006533 and A007678. The difference is that in FR(n) the vertices are not equally spaced around the circle.
Just as in the case of the regular n-gon, when we count the regions in this graph, we may or may not include the regions that lie between the convex hull of the points and the bounding circle.
The first non-simple vertices that do not lie on the y = 0 or x = 0 axes occur for n = 7. If we let A = (sin(3*Pi/14) + cos(Pi/7))/(cos(3*Pi/14) + sin(Pi/7)), and B = (cos(2*Pi/7)+1)/sin(2*Pi/7), then the x coordinate of these vertices is x = +-(A*cos(3*Pi/14) - sin(3*Pi/14) - 1)/(B + A), and their y coordinate is y = -B*x - 1. These values are approximately x = +-0.1930964297 and y = -0.5990311320.

Tom Duff, The Farey Ring graphs FR(2) to FR(10)
Tom Duff, The Farey Ring graph FR(16)
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 7. The two non-simple vertices mentioned in the comments are the two yellow dots in the lower half of the figure on either side of the y axis.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.

Cf. A359117 (regions), A359118 (edges), A359119 (k-gons).

See also A358883, A005728, A006533, A006842, A006843, A007569, A007678.

a(n) = A359118 - A359117 + 1 by Euler's formula.

A359117 Number of regions in the planar Farey Ring graph FR(n) defined in A359116, including the regions between the convex hull and the bounding circle.

Original entry on oeis.org

1, 2, 8, 30, 250, 548, 3180, 7468, 20684, 36164, 112406, 163600, 425144, 636208, 1028934, 1581766, 3323182, 4249976, 8216442, 10668790, 15329216

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Dec 17 2022

Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.

Cf. A359116 (vertices), A359118 (edges), A359119 (k-gons), A358886, A006842, A006843, A005728, A007678.

a(n) = A359118 - A359116 + 1 by Euler's formula.

A364829 a(n) is the number of regions inside a regular n-gon after inscribing (always starting from the same point), for each proper divisor d_i of n greater than 1, a regular d_i-gon.

Original entry on oeis.org

0, 0, 1, 2, 1, 6, 1, 6, 4, 8, 1, 24, 1, 10, 13, 14, 1, 28, 1, 32, 15, 14, 1, 64, 6, 16, 13, 38, 1, 84, 1, 30, 19, 20, 21, 104, 1, 22, 21, 88, 1, 102, 1, 50, 57, 26, 1, 148, 8, 54, 25, 56, 1, 98, 25, 106, 27, 32, 1, 300, 1, 34, 69, 62, 27, 134, 1, 68, 31, 144

Offset: 1

Paolo Xausa, Aug 09 2023

nonn

Inspired by an X (or Twitter) post by Matt Henderson (see links section).

This is A364828 without the outer circle.

Matt Henderson, X post, 2023.
Paolo Xausa, Illustration of first terms.
Paolo Xausa, Illustration of a(60).

Cf. A006600, A007678, A364828, A364830, A364838.

a(n) = 1, if n is prime.
a(n) = n - 2, if n > 1 is a power of 2.
a(n) = A364828(n) - n.

A368813 Number of regions in a regular 2n-gon when all vertices are connect by straight lines except for the n lines joining diametrically opposite vertices.

Original entry on oeis.org

0, 1, 13, 57, 171, 361, 813, 1489, 2215, 4081, 6249, 8329, 13027, 17977, 20221, 32033, 41583, 50545, 66881, 83161, 93871, 124521, 150237, 173233, 213351, 251473, 285445, 342889, 397011, 432121, 524149, 598081, 663103, 769217, 867441, 960121, 1091723, 1218889, 1333489, 1506321, 1667799

Offset: 1

Scott R. Shannon, Jan 06 2024

nonn

Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 15.

Cf. A368814 (vertices), A368815 (edges), A368816 (k-gons), A368755, A007678.

a(n) = A368815(n) - A368814(n) + 1 by Euler's formula.

A160860 The least possible number of pieces resulting from cutting a convex n-gon along all its diagonals.

Original entry on oeis.org

1, 4, 11, 24, 47, 80

Offset: 3

Vladimir Letsko, May 29 2009, May 30 2009, Apr 20 2010

It seems that a(9)=137 and a(n) = A007678(n) for all even n.

Vladimir Letsko, Illustration of all cases for number of sides from 3 to 8
Vladimir Letsko, Illustration of all cases for number of sides from 3 to 8 [Cached copy, pdf version only]
Vladimir Letsko, Proof for n = 7 and n = 8 and example for n = 9 (in Russian)
Vladimir Letsko, Proof for n = 7 and n = 8 and example for n = 9 (in Russian). [Cached copy, pdf version only]
V. A. Letsko, M. A. Voronina, Classification of convex polygons, Grani Poznaniya, 1(11), 2011. (in Russian)
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon.

Cf. A006522, A007678, A230281.

A187781 Number of noncongruent polygonal regions in a regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 1, 3, 3, 7, 7, 14, 14, 25, 21, 41, 40, 63, 60, 92, 72, 129, 121, 175, 166, 231, 192, 298, 285, 377, 360, 469, 350, 575, 553, 696, 666, 833, 744, 987, 956, 1159, 1123, 1350, 1165, 1561, 1508, 1793, 1741, 2047, 1875, 2324, 2255, 2625, 2563, 2951, 2761, 3303, 3214, 3682, 3588, 4089, 3695

Offset: 3

Martin Renner, Jan 05 2013

			a(5) = 3 since the 11 regions of a regular pentagon with all diagonals drawn consist of three different noncongruent polygons: two different triangles (each 5 times) and 1 pentagon.
a(6) = 3 since the 24 regions of the regular hexagon with all diagonals drawn consist of three different noncongruent polygons: 2 triangles (one 6 times, one 12 times) and 1 quadrilateral (6 times).
a(7) = 7 since the 50 regions of the regular heptagon with all diagonals drawn consist of seven different noncongruent polygons: 4 triangles (three 7 times, one 14 times), 1 quadrilateral (7 times), 1 pentagon (7 times) and 1 heptagon.

Sascha Kurz, Anzahl von Dreiecken eines regelmäßigen n-Ecks.
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11 (1998), nr. 1, pp. 135-156; doi: 10.1137/S0895480195281246; arXiv: math.MG/9508209.
Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals.
Index to sequences on drawing diagonals in regular polygons.

Cf. A006561, A007678, A165217, A187782.

Corrected a(12) and a(16), extended from a(18) through a(60), corrected small typo in a(7) example - Christopher Scussel, Jun 23 2023

cp's OEIS Frontend

A341877 The number of regions inside a vesica piscis 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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A344907 Number of polygon edges formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line.

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A345649 Number of vertices formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

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A345650 Number of polygon edges formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

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A359116 Mark the points of the Farey series F_n on a strip of paper and wrap it around a circle of circumference 1 so the endpoints 0 and 1 coincide; draw a chord between every pair of the Farey points; a(n) is the number of vertices in the resulting graph.

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A359117 Number of regions in the planar Farey Ring graph FR(n) defined in A359116, including the regions between the convex hull and the bounding circle.

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A364829 a(n) is the number of regions inside a regular n-gon after inscribing (always starting from the same point), for each proper divisor d_i of n greater than 1, a regular d_i-gon.

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A368813 Number of regions in a regular 2n-gon when all vertices are connect by straight lines except for the n lines joining diametrically opposite vertices.

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A160860 The least possible number of pieces resulting from cutting a convex n-gon along all its diagonals.

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A187781 Number of noncongruent polygonal regions in a regular n-gon with all diagonals drawn.

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