A007569 -id:A007569 - cp's OEIS frontend

A292104 Number of interior points that are the intersections of exactly two chords in the configuration A006561(n).

0, 0, 0, 1, 5, 12, 35, 40, 126, 140, 330, 228, 715, 644, 1365, 1168, 2380, 1512, 3876, 3360, 5985, 5280, 8855, 6144, 12650, 11284, 17550, 15680, 23751, 13800, 31465, 28448, 40920, 37264, 52360, 42444, 66045, 60648, 82251, 75720, 101270, 75012, 123410, 114400, 148995, 138644, 178365, 152064

Offset: 1

N. J. A. Sloane, Sep 14 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..120 from Scott R. Shannon)
N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]

Cf. A006561. Column k=2 of A292105.

a(31)-a(48) from Scott R. Shannon, Mar 04 2022

A292105 Irregular triangle read by rows: T(n,k) = the number of interior points that are the intersections of exactly k chords in the configuration A006561(n) (n >= 1, k >= 1).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 5, 0, 12, 1, 0, 35, 0, 40, 8, 1, 0, 126, 0, 140, 20, 0, 1, 0, 330, 0, 228, 60, 12, 0, 1, 0, 715, 0, 644, 112, 0, 0, 0, 1, 0, 1365, 0, 1168, 208, 0, 0, 0, 0, 1, 0, 2380, 0, 1512, 216, 54, 54, 0, 0, 0, 1, 0, 3876, 0, 3360, 480, 0, 0, 0, 0, 0, 0, 1, 0, 5985

Offset: 1

N. J. A. Sloane, Sep 14 2017

			Triangle begins:
  0;
  0;
  0;
  0,   1;
  0,   5;
  0,  12,  1;
  0,  35;
  0,  40,  8,  1;
  0, 126;
  0, 140, 20,  0, 1;
  0, 330;
  0, 228, 60, 12, 0, 1;
See the attached text file for the first 100 rows.

Seiichi Manyama, Rows n = 1..250, flattened
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 1995-2006, which has fewer typos than the SIAM version.
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 (1998). [Copy on SIAM web site]
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). [Copy on B. Poonen's web site]
B. Poonen and M. Rubinstein, Mathematica programs for A006561 and related sequences
N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)
Scott R. Shannon, Table for n=1..100.
Scott R. Shannon, Image of 8-gon.
Scott R. Shannon, Image of 9-gon.
Scott R. Shannon, Image of 12-gon.

Columns give A292104, A101363 (2n-gon), A101364, A101365.
Row sums give A006561.
Cf. A335102.

a(27) and beyond by Scott R. Shannon, May 15 2022

A331449 a(0) = 1 by convention; for n>0, a(n) is the number of points in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

Original entry on oeis.org

1, 5, 37, 257, 817, 2757, 4825, 12293, 19241, 33549, 49577, 87685, 101981, 178465, 220113, 286357, 379097, 551669, 606241, 880293, 951445, 1209049, 1507521, 1947877, 2002669, 2624409, 3056093, 3550425, 3955049, 5069037, 5062153, 6669665, 7081969, 8143193, 9365089, 10296469

Offset: 0

N. J. A. Sloane, Jan 23 2020

nonn

Equivalently, this is A334690(n) + 4*n.

Zhao Hui Du, Table of n, a(n) for n = 0..136 (terms 0..52 from Lars Blomberg).
Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 20.
Scott R. Shannon, Image for n = 1.
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 = 10.

Cf. A255011, A331448.

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

By Euler's formula, a(n) = A331448(n) - A255011(n) + 1.

a(11)-a(35) from Giovanni Resta, Jan 28 2020

A330846 The number of vertices inside a hexagon 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

19, 301, 1753, 6913, 17713, 38497, 80473, 139927, 225595, 356329, 549967, 756901, 1113283, 1474867, 1913365, 2620789, 3418081, 4119175, 5428993, 6453235, 7861957, 9761335, 11913115, 13695055, 16489633, 19439797, 22464883, 25997329, 30860593, 33866305

Offset: 1

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

nonn

See the links in A331931 for images of the hexagons.

Cf. A331931 (regions), A331932 (n-gons), A330845 (edges), A092866, A332599, A007569.

a(9)-a(30) from Lars Blomberg, May 12 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

A371373 Place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. The sequence gives the total number of vertices formed.

Original entry on oeis.org

1, 2, 4, 4, 25, 19, 140, 144, 460, 500, 1210, 901, 2587, 2758, 4696, 5136, 8687, 7831, 14136, 14600, 21610, 22572, 32246, 31033, 46125, 47450, 63748, 65772, 86565, 82051, 114824, 117760, 148930, 152796, 190820, 189973, 241055, 247038, 300028, 306840, 369943, 367711, 451586, 459448

Offset: 1

Scott R. Shannon, Mar 20 2024

nonn

The number of circles that cross to form the intersections follows a similar pattern to that seen in A371254; see that sequence for further information. The details of the crossing counts are given in A371377.

Scott R. Shannon, Image for n = 3. In this and other images the initial circle on which the n points are placed is drawn in a thicker white line for clarity.
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.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 12. Note the 6 circle intersections shown in blue.
Scott R. Shannon, Image for n = 15. Note the 5 circle intersections shown in green.
Scott R. Shannon, Image for n = 20.
Scott R. Shannon, Image for n = 24.
Scott R. Shannon, Image for n = 30. Note the 9 circle intersections shown in violet.

Cf. A371374 (regions), A371375 (edges), A371376 (k-gons), A371377 (vertex crossings), A371254, A007569, A358746, A331702.

a(n) = A371375(n) - A371374(n) + 1 by Euler's formula.

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

A371254 Number of vertices formed when n equally spaced points are placed around a circle and all pairs of points are joined by an interior arc whose radius equals the circle's radius.

Original entry on oeis.org

1, 2, 4, 4, 15, 7, 70, 64, 208, 220, 550, 397, 1131, 1162, 1981, 2128, 3723, 3259, 5966, 6000, 9010, 9240, 13524, 12745, 19325, 19266, 26434, 26684, 35931, 33301, 47368, 47616, 61216, 61676, 78330, 76789, 98901, 99674, 122656, 123560

Offset: 1

Scott R. Shannon, Mar 16 2024

nonn

Other than for n = 3, 4, and 6, all graphs so far investigated in this sequence contain some internal vertices which are created from the intersections of both 2 and 3 arcs, i.e., no graph contains only simple intersections. This is in contrast to the case where the point pairs are connected by straight lines, see A007569 and A335102, where the odd-n graphs contain only simple intersections. See the attached images.
Other patterns for the intersection arc counts are also seen. If n is divisible by 3 then a central vertex is always present that is created from the crossing of n arcs. If n is divisible by 6, then internal vertices are present that are created from the crossing of 6 arcs. For n = 15 and n = 45, internal vertices are present that are created from the crossing of 5 arcs - it is likely all graphs with n = 15+30*k, k>=0, contain such vertices.
For n = 30, the graph also contains internal vertices that are created from the crossing of 9 arcs. It is likely that all graphs with n divisible by 30 contain such vertices. As the graphs created from the straight line diagonal intersections of the regular n-gon, see A007569, also have the maximum possible line intersection count of 7 when n is divisible by 30, it is plausible that 9 is the maximum possible arc intersection count for any internal vertex, other than the central vertex when n is divisible by 3.
Assuming these patterns hold for all n, is it possible that there is a general formula for the number of vertices, analogous to that in A007569 for the intersections of chords in a regular n-gon?

B. Poonen and M. Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, arXiv:math/9508209v3 [math.MG], 1995-2006.
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.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 12.
Scott R. Shannon, Image for n = 15. Note the 5 arc intersections shown in green.
Scott R. Shannon, Image for n = 20.
Scott R. Shannon, Image for n = 24.
Scott R. Shannon, Image for n = 30. Note the 9 arc intersections shown in violet.

Cf. A371253 (regions), A371255 (edges), A371274 (k-gons), A371264 (vertex crossings), A370980 (number of circles), A371373 (complete circles), A007569, A335102, A358746, A331702, A359252.

a(n) = A371255(n) - A371253(n) + 1 by Euler's formula.

A333035 Number of vertices in an equal-armed cross with arms of length n (see Comments in A331456 for definition).

Original entry on oeis.org

5, 69, 397, 1417, 3717, 8009, 15361, 26777, 43697, 67597, 100245, 143177, 199009, 269197, 356789, 464577, 595521, 751129, 935825, 1151881, 1403953, 1695765, 2031485, 2413337, 2848373, 3337781, 3888917, 4505277, 5191557, 5952313, 6796713, 7725945, 8747121

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Mar 06 2020

nonn

See the links in A331456 for images of the crosses.

Lars Blomberg, Table of n, a(n) for n = 0..48

Cf. A331456 (regions), A333036 (edges), A333037 (n-gons), A092866, A332599, A007569.

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

A330847 The number of vertices inside a pentagon 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

10, 121, 1055, 3506, 10410, 21111, 43740, 74526, 124490, 190291, 288190, 403321, 573805, 771191, 1027550, 1331896, 1721765, 2157301, 2712175, 3322441, 4067325, 4904491, 5900185, 6975151, 8268755, 9678566, 11297345, 13060231, 15102955, 17263311, 19784995

Offset: 1

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

nonn

See the links in A331929 for images of the pentagons.

Lars Blomberg, Table of n, a(n) for n = 1..35

Cf. A331929 (regions), A331939 (n-gons), A329710 (edges), A330846, A092866, A332599, A007569

a(9) and beyond from Lars Blomberg, May 11 2020

cp's OEIS Frontend

A292104 Number of interior points that are the intersections of exactly two chords in the configuration A006561(n).

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A292105 Irregular triangle read by rows: T(n,k) = the number of interior points that are the intersections of exactly k chords in the configuration A006561(n) (n >= 1, k >= 1).

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A331449 a(0) = 1 by convention; for n>0, a(n) is the number of points in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

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A330846 The number of vertices inside a hexagon 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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A371373 Place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. The sequence gives the total number of vertices formed.

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Author

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Comments

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Formula

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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Author

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Comments

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Crossrefs

Extensions

A371254 Number of vertices formed when n equally spaced points are placed around a circle and all pairs of points are joined by an interior arc whose radius equals the circle's radius.

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Comments

Links

Crossrefs

Formula

A333035 Number of vertices in an equal-armed cross with arms of length n (see Comments in A331456 for definition).

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A330847 The number of vertices inside a pentagon 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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