A092098 -id:A092098 - cp's OEIS frontend

A092867 Number of regions in an equilateral triangular figure formed by the straight line segments connecting all vertices and all points that divide the sides into n equal parts.

1, 12, 75, 252, 715, 1572, 3109, 5676, 9291, 14556, 22081, 32502, 44935, 62868, 83286, 108384, 140152, 181710, 225565, 282978, 342792, 415614, 502318, 606642, 708505, 839874, 983007, 1141416, 1315102, 1529526, 1733476, 1994550, 2259420, 2559990, 2878053, 3237414, 3593521, 4047906, 4510590, 5002350, 5506918, 6128100, 6704800, 7414518, 8113992, 8858622, 9682927, 10626774, 11478142, 12519492

Offset: 1

Hugo Pfoertner, Mar 15 2004

			a(2)=12 because the 6 line segments mutually connecting the vertices and the mid-side nodes form 12 congruent right triangles of two different sizes.
a(3)=75: 48 triangles, 24 quadrilaterals and 3 pentagons are formed. See pictures at Pfoertner link.

Hugo Pfoertner, Intersections of diagonals in polygons of triangular shape.
Cynthia Miaina Rasamimanananivo and Max Alekseyev, Sage program for this sequence
Scott R. Shannon, Triangle regions for n = 2.
Scott R. Shannon, Triangle regions for n = 3.
Scott R. Shannon, Triangle regions for n = 4.
Scott R. Shannon, Triangle regions for n = 5.
Scott R. Shannon, Triangle regions for n = 6.
Scott R. Shannon, Triangle regions for n = 7.
Scott R. Shannon, Triangle regions for n = 8.
Scott R. Shannon, Triangle regions for n = 9.
Scott R. Shannon, Triangle regions for n = 10.
Scott R. Shannon, Triangle regions for n = 11.
Scott R. Shannon, Triangle regions for n = 12.
Scott R. Shannon, Triangle regions for n = 13.
Scott R. Shannon, Triangle regions for n = 14.
Scott R. Shannon, Triangle regions for n = 9, random distance-based coloring.
Scott R. Shannon, Triangle regions for n = 12, random distance-based coloring
Index entries for sequences related to stained glass windows
Sequences formed by drawing all diagonals in regular polygon

Cf. A092866 (number of intersections),  A274585 (number of points both inside and on the triangle sides), A274586 (number of edges), A331911 (number of n-gons).
Cf. A092098 (regions in triangle cut by line segments connecting vertices with subdivision points on opposite side), A006533 (regions formed by all diagonals in regular n-gon), A002717 (triangles in triangular matchstick arrangement).
If the boundary points are in general position, we get A367117, A213827, A367118, A367119. - N. J. A. Sloane, Nov 09 2023

By the Euler characteristic, a(n) = A274586(n) - A274585(n) + 1 = A274586(n) - A092866(n) - 3n - 1.

a(1)=1 prepended by Max Alekseyev, Jun 29 2016
a(6)-a(50) from Cynthia Miaina Rasamimanananivo, Jun 28 2016, Jul 01 2016, Aug 05 2016, Aug 15 2016
Definition edited by N. J. A. Sloane, May 13 2020

A255011 Number of polygons formed by connecting all the 4n points on the perimeter of an n X n square by straight lines; a(0) = 0 by convention.

Original entry on oeis.org

0, 4, 56, 340, 1120, 3264, 6264, 13968, 22904, 38748, 58256, 95656, 120960, 192636, 246824, 323560, 425408, 587964, 682296, 932996, 1061232, 1327524, 1634488, 2049704, 2227672, 2806036, 3275800, 3810088, 4307520, 5298768, 5577096, 6958848, 7586496, 8672520, 9901352

Offset: 0

Johan Westin, Feb 12 2015

nonn

There are n+1 points on each side of the square, but that counts the four corners twice, so there are a total of 4n points on the perimeter. - N. J. A. Sloane, Jan 23 2020
a(n) is always divisible by 4, by symmetry. If n is odd, a(n) is divisible by 8.
From Michael De Vlieger, Feb 19-20 2015: (Start)
For n > 0, the vertices of the bounding square generate diametrical bisectors that cross at the center. Thus each diagram has fourfold symmetry.
For n > 0, an orthogonal n X n grid is produced by corresponding horizontal and vertical points on opposite sides.
Terms {1, 3, 9} are not congruent to 0 (mod 8).
Number of edges: {0, 8, 92, 596, 1936, 6020, 11088, 26260, 42144, 72296, 107832, ...}. See A331448. (End)

			For n = 3, the perimeter of the square contains 12 points:
  * * * *
  *     *
  *     *
  * * * *
Connect each point to every other point with a straight line inside the square. Then count the polygons (or regions) that have formed. There are 340 polygons, so a(3) = 340.
For n = 1, the full picture is:
  *-*
  |X|
  *-*
The lines form four triangular regions, so a(1) = 4.
For n = 0, the square can be regarded as consisting of a single point, producing no lines or polygons, and so a(0) = 0.

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 on arXiv, arXiv:2009.07918 [math.CO], 2020.
Michael De Vlieger, Diagrams of A255011(n) for n <= 10
B. Poonen and M. Rubinstein (1998) The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11(1), pp. 135-156, doi:10.1137/S0895480195281246, arXiv:math.MG/9508209 (has fewer typos than the SIAM version)
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, 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.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 20.

Cf. A092098 (triangular analog), A331448 (edges), A331449 (points), A334699 (k-gons).

For the circular analog see A006533, A007678.

No formula is presently known. - N. J. A. Sloane, Feb 04 2020

a(11)-a(29) from Hiroaki Yamanouchi, Feb 23 2015

Offset changed by N. J. A. Sloane, Jan 23 2020

A091908 Number of interior intersection points made by the straight line segments connecting the edges of an equilateral triangle with the n-1 points resulting from a subdivision of the sides into n equal pieces, counting coinciding intersection points only once.

Original entry on oeis.org

0, 1, 12, 13, 48, 49, 108, 109, 192, 193, 300, 301, 432, 433, 576, 589, 768, 769, 972, 961, 1200, 1201, 1452, 1405, 1728, 1729, 2028, 2029, 2352, 2341, 2700, 2701, 3072, 3073, 3444, 3469, 3888, 3889, 4332, 4297, 4800, 4777, 5292, 5293, 5724, 5809, 6348

Offset: 1

Hugo Pfoertner, Feb 19 2004

nonn

In a drawing the distinction between simple and multiple intersection points may be difficult due to near-coincidences. E.g. there are no coincident intersections for n=7.
Note that 3 divides a(2k)-1 and a(2k+1). - T. D. Noe, Jun 29 2005
The interior intersection points only can be the result of the concurrency of 2 or 3 segments by construction. It is easy to see that the total number of 2-intersections N2 is 3*(n-1)^2 (which includes every 3-intersection as two 2-intersections) by symmetry. But we are interested in excluding the concurrency of more than 2. By Ceva's theorem necessary and sufficient condition for 3 concurrent segments that connect the edges with the opposite side, the number of 3-intersections N3 is the same as the number of (i,j,k) belonging to [1,n-1]x[1,n-1]x[1,n-1] such that (i/(n-i))*(j/(n-j))*(k/(n-k))=1. Thus the terms a(n)=N2-2*N3. - Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 26 2006
If n is even then a(n) < 3*(n-1)^2; if n is odd then a(n) = 3*(n-1)^2 except for n in A332378. - N. J. A. Sloane, Feb 14 2020

			a(3)=12 because the 3*2 line segments intersect each other in 12 distinct points (see pictures given at link)
a(4)=13 because the 27 intersections form 6 two line intersection points and 7 three line intersection points.

Hugo Pfoertner, Table of n, a(n) for n = 1..1000
Hugo Pfoertner, Visualization of diagonal intersections in an equilateral triangle.
Index entries for sequences formed by drawing all diagonals in regular polygon

Cf. A091910 = radial locations of intersection points, A092098 = number of regions that the line segments cut the triangle into, A006561.

For the basic properties of the underlying graph, see A092098 (cells), A331782 (vertices), A331782 (vertices), A332376 & A332377 (edges). - N. J. A. Sloane, Feb 14 2020

for(n=1,70,conc=0;for(i=1,n-1,for(j=1,n-1,for(k=1,n-1,if(i*j*k/((n-i)*(n-j)*(n-k))==1,conc++))));print1(3*(n-1)^2-2*conc,",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 26 2006

More terms from T. D. Noe, Jun 29 2005

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 26 2006

A331782 Total number of vertices in graph formed by the straight line segments connecting the edges of an equilateral triangle with the n-1 points resulting from a subdivision of the sides into n equal pieces, counting coinciding intersection points only once.

Original entry on oeis.org

3, 7, 21, 25, 63, 67, 129, 133, 219, 223, 333, 337, 471, 475, 621, 637, 819, 823, 1029, 1021, 1263, 1267, 1521, 1477, 1803, 1807, 2109, 2113, 2439, 2431, 2793, 2797, 3171, 3175, 3549, 3577, 3999, 4003, 4449, 4417, 4923, 4903, 5421, 5425, 5859, 5947, 6489, 6397, 7059, 7063, 7653, 7657, 8271, 8275, 8889

Offset: 1

N. J. A. Sloane, Feb 13 2020

nonn

a(n) <= 3*(n^2-n+1), with equality iff n is odd and not a member of A332378. A331423 gives the difference between a(n) and the upper bound.

N. J. A. Sloane, Table of n, a(n) for n = 1..1000

Cf. A091908, A092098 (number of cells), A332376 (number of edges), A332378, A331423.

a(n) = A091908(n) + 3*n.

A108914 Number of regions formed inside square by diagonals and the segments joining the vertices to the points dividing the sides into n equal length segments.

Original entry on oeis.org

4, 32, 96, 188, 332, 460, 712, 916, 1204, 1488, 1904, 2108, 2716, 3080, 3532, 4068, 4772, 5140, 6016, 6392, 7188, 7992, 8936, 9260, 10484, 11312, 12208, 12968, 14396, 14660, 16504, 17220, 18436, 19680, 20756, 21548, 23692, 24728, 25992, 26868, 29204, 29704, 32176, 33068, 34444, 36552, 38552

Offset: 1

Len Smiley and Brian Wick ( mathclub(AT)math.uaa.alaska.edu ), Jul 19 2005

nonn

Scott R. Shannon, Table of n, a(n) for n = 1..100
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 = 11.
Scott R. Shannon, Image for n = 30.
L. Smiley, The case n=6. Note 3- and 4-fold off-diagonal concurrencies.
L. Smiley, The case n=7. Note there are no off-diagonal concurrencies.

A092098 is the corresponding count for triangles.

A355949 (vertices), A355948 (edges), A355992 (k-gons), A355838, A355798.

If n=1 or n is prime, a(n)=18*n^2-26*n+12.
If n is composite, vanishing regions from 3- and 4-fold concurrency must be subtracted.
a(n) = A355948(n) - A355949(n) + 1 by Euler's formula.

a(23), a(33) corrected, a(41) and above by Scott R. Shannon, Jul 22 2022

A345459 Number of polygons formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

Original entry on oeis.org

0, 4, 80, 568, 2024, 6052, 12144, 26976, 45024, 76724, 116840, 191128, 245976, 388452, 501888, 661476, 870168, 1199724, 1402096, 1911384, 2188320, 2739280, 3371264, 4224288, 4617224, 5801372, 6780568

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Jun 20 2021

The width/height of the entire figure grows as ~ 2*n^3 for large n. See the Formula section below.

			a(2) = 80. Connecting the 8 perimeter points results in the creation of forty-eight 3-gons and eight 4-gons inside the square while creating twenty-four 3-gons outside the square, giving eighty polygons in total. See the linked images.

Scott R. Shannon, Image for n = 2. In this and other images the square's points are highlighted as white dots while the outer open regions, which are not counted, are darkened. The key for the edge-number coloring is shown at the top-left of the image.
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.

Cf. A255011 (number inside the square), A345648 (number outside the square), A345649 (number of vertices), A345650 (number of edges), A344993, A344857, A092098, A007678.

a(n) = A345650(n) - A345649(n) + 1.

Assuming the square is of size n x n centered on the origin the x (or y) offset for the eight outermost vertices is n^3 - 2*n^2 + 3*n/2, which have a corresponding y (or x) offset of n^2 - 3*n/2 + 1. The total distance from the origin of these vertices is sqrt(n^6 - 4*n^5 + 8*n^4 - 9*n^3 + 13*n^2/2 - 3*n + 1).

A365929 Number of intersections formed within a triangle by placing n points "in general position" on each of the three sides and connecting each point to each of the points on the other two sides using straight lines.

Original entry on oeis.org

0, 0, 15, 108, 396, 1050, 2295, 4410, 7728, 12636, 19575, 29040, 41580, 57798, 78351, 103950, 135360, 173400, 218943, 272916, 336300, 410130, 495495, 593538, 705456, 832500, 975975, 1137240, 1317708, 1518846, 1742175, 1989270, 2261760, 2561328, 2889711, 3248700, 3640140, 4065930, 4528023

Offset: 0

Vijay Srinivas Balaji, Sep 23 2023

There are n points on each of the three sides (not counting the vertices of the triangle). Each point must be connected to every point on the other two sides. A033428(n) = 3*n^2 gives the number of lines.
Comments from N. J. A. Sloane, Oct 29 2023: (Start)
"In general position" means that all interior intersection points are simple. No three-way or higher intersections are permitted.
If the 3*n+3 boundary points are included in the count, there are 3*A366478 points.
For the configurations where the boundary points are equally spaced and every pair of boundary points is joined by a chord, see A091908, A092098, A331782.
(End)

			a(5) = (3/4) * 5^2 * (3*5^2 - 4*5 + 1) = 1050.

Vijay Srinivas Balaji, Formulating A Conjecture For Intersections Created From Crossing Lines Within Different Polygons, International School of Helsingborg, 2023.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Vijay Srinivas Balaji, Diagram of Intersections for a Triangle.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 10.

Cf. A367015 (number of regions), A366932 (number of edges), A366478 (vertices including boundary points), A033428 (number of chords).

A332376 Total number of edges in graph formed by the straight line segments connecting the edges of an equilateral triangle with the n-1 points resulting from a subdivision of the sides into n equal pieces.

Original entry on oeis.org

3, 12, 39, 54, 123, 144, 255, 282, 435, 468, 663, 702, 939, 984, 1245, 1314, 1635, 1692, 2055, 2100, 2523, 2592, 3039, 3042, 3603, 3684, 4215, 4302, 4875, 4950, 5583, 5682, 6339, 6444, 7107, 7254, 7995, 8112, 8895, 8964, 9843, 9936, 10839, 10974

Offset: 1

N. J. A. Sloane, Feb 13 2020

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..1000

Cf. A091908, A092098 (number of cells), A331782 (number of vertices).

Equals three times A332377.

a(n) = A092098(n) + A331782(n) - 1 (Euler's formula).

A367117 Place n points in general position on each side of an equilateral triangle, and join every pair of the 3*n+3 boundary points by a chord; sequence gives number of vertices in the resulting planar graph.

Original entry on oeis.org

3, 12, 72, 282, 795, 1818, 3612, 6492, 10827, 17040, 25608, 37062, 51987, 71022, 94860, 124248, 159987, 202932, 253992, 314130, 384363, 465762, 559452, 666612, 788475, 926328, 1081512, 1255422, 1449507, 1665270, 1904268, 2168112, 2458467, 2777052, 3125640, 3506058, 3920187, 4369962

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Nov 05 2023

nonn

"In general position" implies that the internal lines (or chords) only have simple intersections. There is no interior point where three or more chords meet.

Note that although the number of k-gons in the graph will vary as the edge points change position, the total number of regions will stay constant as long as all internal vertices remain simple.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 5.

Cf. A367118 (regions), A367119 (edges).
See also A091908, A092098, A331782, A365929.
If the boundary points are equally spaced, we get A274585, A092866, A274586, A092867. - N. J. A. Sloane, Nov 09 2023

A367117[n_]:=3/4(n+1)(3n^3+n^2+4);Array[A367117,50,0] (* Paolo Xausa, Nov 09 2023 *)

Theorem: a(n) = (3/4)*(n+1)*(3*n^3+n^2+4).

a(n) = A367119(n) - A367118(n) + 1 by Euler's formula.

A367118 Place n points in general position on each side of an equilateral triangle, and join every pair of the 3*n+3 boundary points by a chord; sequence gives number of regions in the resulting planar graph.

Original entry on oeis.org

1, 13, 82, 307, 841, 1891, 3718, 6637, 11017, 17281, 25906, 37423, 52417, 71527, 95446, 124921, 160753, 203797, 254962, 315211, 385561, 467083, 560902, 668197, 790201, 928201, 1083538, 1257607, 1451857, 1667791, 1906966, 2170993, 2461537, 2780317, 3129106, 3509731, 3924073, 4374067

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Nov 05 2023

nonn

"In general position" implies that the internal lines (or chords) only have simple intersections. There is no interior point where three or more chords meet.

Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 5.

Cf. A367117 (vertices), A367119 (edges), A091908, A092098, A331782, A367015.

If the boundary points are equally spaced, we get A274585, A092866, A274586, A092867. - N. J. A. Sloane, Nov 09 2023

Conjecture: a(n) = (1/4)*(9*n^4 + 12*n^3 + 15*n^2 + 12*n + 4).

a(n) = A367119(n) - A367117(n) + 1 by Euler's formula.

cp's OEIS Frontend

A092867 Number of regions in an equilateral triangular figure formed by the straight line segments connecting all vertices and all points that divide the sides into n equal parts.

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A255011 Number of polygons formed by connecting all the 4n points on the perimeter of an n X n square by straight lines; a(0) = 0 by convention.

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A091908 Number of interior intersection points made by the straight line segments connecting the edges of an equilateral triangle with the n-1 points resulting from a subdivision of the sides into n equal pieces, counting coinciding intersection points only once.

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A331782 Total number of vertices in graph formed by the straight line segments connecting the edges of an equilateral triangle with the n-1 points resulting from a subdivision of the sides into n equal pieces, counting coinciding intersection points only once.

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A108914 Number of regions formed inside square by diagonals and the segments joining the vertices to the points dividing the sides into n equal length segments.

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A345459 Number of polygons formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

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A365929 Number of intersections formed within a triangle by placing n points "in general position" on each of the three sides and connecting each point to each of the points on the other two sides using straight lines.

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A332376 Total number of edges in graph formed by the straight line segments connecting the edges of an equilateral triangle with the n-1 points resulting from a subdivision of the sides into n equal pieces.

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A367117 Place n points in general position on each side of an equilateral triangle, and join every pair of the 3*n+3 boundary points by a chord; sequence gives number of vertices in the resulting planar graph.

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