A331782 -id:A331782 - cp's OEIS frontend

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.

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

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.

A367119 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 edges in the resulting planar graph.

Original entry on oeis.org

3, 24, 153, 588, 1635, 3708, 7329, 13128, 21843, 34320, 51513, 74484, 104403, 142548, 190305, 249168, 320739, 406728, 508953, 629340, 769923, 932844, 1120353, 1334808, 1578675, 1854528, 2165049, 2513028, 2901363, 3333060, 3811233, 4339104, 4920003, 5557368, 6254745, 7015788

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.

See A367117 and A367118 for images of the triangle.

Cf. A367117 (vertices), A367118 (regions), A091908, A092098, A331782, A366932.

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

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

a(n) = A367117 (n) + A367118 (n) - 1 by Euler's formula.

A335411 a(n) is the number of vertices formed by n-secting the angles of an equilateral triangle.

Original entry on oeis.org

3, 7, 21, 25, 63, 67, 129, 133, 219, 199, 333, 337, 471, 475, 633, 637, 819, 823, 1029, 1009, 1263, 1267, 1521, 1525, 1803, 1807, 2109, 2113, 2439, 2419, 2793, 2797, 3171, 3175, 3573, 3577, 3999, 4003, 4449, 4429, 4923, 4927, 5421, 5425, 5943, 5947, 6489

Offset: 1

Lars Blomberg, Jun 08 2020

nonn

See A277402 for illustrations.

Lars Blomberg, Table of n, a(n) for n = 1..500

Cf. A331782, A277402 (regions), A335412 (edges), A335413 (ngons).

Empirically for 12 < n < 500: a(n) = a(n-2) + a(n-10) - a(n-12) + 120.
Conjectures from Colin Barker, Jun 08 2020: (Start)
G.f.: x*(3 + 4*x + 11*x^2 + 24*x^4 + 24*x^6 + 24*x^8 - 24*x^9 + 45*x^10 + 20*x^11 - 11*x^12) / ((1 - x)^3*(1 + x)^2*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-10) - a(n-11) - a(n-12) + a(n-13) for n>13.
(End)
Colin Barker's recurrence conjecture holds for 13 < n <= 500. Lars Blomberg, Jun 12 2020
Empirical: a(2*k - 1) = 3*(4*k^2 - 6*k + 3), for k >= 1. - Ivan N. Ianakiev, Jul 15 2020

A332377 One-third of 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

1, 4, 13, 18, 41, 48, 85, 94, 145, 156, 221, 234, 313, 328, 415, 438, 545, 564, 685, 700, 841, 864, 1013, 1014, 1201, 1228, 1405, 1434, 1625, 1650, 1861, 1894, 2113, 2148, 2369, 2418, 2665, 2704, 2965, 2988, 3281, 3312, 3613, 3658, 3919, 4008, 4325

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), A332376.

a(n) = (A092098(n) + A331782(n) - 1)/3 = A332376(n)/3.

A354544 Table read by antidiagonals: T(n,k) (n >= 3, k >= 1) is the number of vertices formed in a regular n-gon by straight line segments when connecting the n corner vertices to the points dividing the sides into k equal parts.

Original entry on oeis.org

3, 7, 5, 21, 25, 10, 25, 81, 61, 19, 63, 157, 285, 205, 42, 67, 301, 476, 541, 358, 57, 129, 381, 1020, 1327, 1526, 681, 135, 133, 665, 1311, 2185, 2682, 2417, 1234, 171, 219, 821, 2215, 3067, 5250, 5073, 4716, 2131, 341, 223, 1109, 2666, 4921, 7246, 8937, 8623, 6861, 3169, 313

Offset: 3

Scott R. Shannon, Aug 18 2022

			The table begins:
3,    7,     21,     25,     63,     67,     129,    133,     219,     223,...
5,    25,    81,     157,    301,    381,    665,    821,     1109,    1353,...
10,   61,    285,    476,    1020,   1311,   2215,   2666,    3810,    4421,...
19,   205,   541,    1327,   2185,   3067,   4921,   6739,    8401,    10507,...
42,   358,   1526,   2682,   5250,   7246,   11214,  14050,   19418,   23094,...
57,   681,   2417,   5073,   8937,   13089,  19473,  26049,   33769,   42497,...
135,  1234,  4716,   8623,   16173,  22780,  34398,  43813,   59391,   71614,...
171,  2131,  6861,   14271,  24731,  36161,  53071,  70751,   91761,   115001,...
341,  3169,  11451,  21143,  38665,  55221,  81983,  105403,  141405,  171689,...
313,  4837,  14653,  31009,  53989,  78589,  115909, 154105,  199777,  249961,...
728,  6787,  23491,  43850,  78858,  113517, 166829, 215788,  287404,  350663,...
771,  9927,  30479,  61951,  105155, 157851, 224043, 299727,  386807,  485367,...
1380, 12856, 43080,  81136,  144180, 208636, 304500, 395536,  524040,  641656,...
1393, 17633, 54001,  109265, 184785, 277745, 392737, 525169,  677729,  849249,...
2397, 22288, 73066,  138177, 243355, 353686, 513264, 668815,  882793,  1083564,...
1855, 27595, 88291,  177085, 302167, 450469, 641539, 855829,  1106119, 1384183,...
3895, 36139, 116337, 220933, 386403, 563351, 814093, 1063393, 1399407, 1721059,...
.
.
See the attached text file for more examples and the cross references for further images.

Scott R. Shannon, Image of T(4,9) = 1109.
Scott R. Shannon, Image of T(5,7) = 2215.
Scott R. Shannon, Image of T(7,6) = 7246.
Scott R. Shannon, Image of T(10,3) = 6861.
Scott R. Shannon, Image of T(11,4) = 21143.
Scott R. Shannon, Table for n=3..25.

Cf. A356044 (number of regions), A007569 (first column), A331782 (first row), A355949 (second row).

T(n,1) = A007569(n).
T(3,k) = A331782(k).
T(4,k) = A355949(k).

A356119 Irregular table read by rows: T(n,k) is the number of k-sided polygons formed, for k>=3, in an equilateral triangle when straight line segments connect the three corner vertices to the points dividing the sides into n equal parts.

Original entry on oeis.org

1, 6, 12, 3, 3, 1, 24, 6, 36, 9, 9, 7, 48, 24, 6, 72, 21, 15, 19, 84, 48, 12, 6, 108, 51, 33, 25, 132, 78, 18, 18, 168, 69, 51, 43, 180, 120, 48, 18, 216, 135, 57, 61, 252, 156, 66, 36, 294, 159, 105, 67, 312, 234, 84, 48, 372, 225, 117, 103, 408, 264, 138, 60, 456, 291, 159, 121, 486, 372, 138, 84

Offset: 1

Scott R. Shannon, Jul 27 2022

The maximum sided k-gon up to n = 250 is the 6-gon; it is likely this is the maximum sided k-gon for all n.

See A092098 for more images of the triangle.

			The table begins:
1;
6;
12,  3,   3,   1;
24,  6;
36,  9,   9,   7;
48,  24,  6;
72,  21,  15,  19;
84,  48,  12,  6;
108, 51,  33,  25;
132, 78,  18,  18;
168, 69,  51,  43;
180, 120, 48,  18;
216, 135, 57,  61;
252, 156, 66,  36;
294, 159, 105, 67;
312, 234, 84,  48;
372, 225, 117, 103;
408, 264, 138, 60;
456, 291, 159, 121;
486, 372, 138, 84;
.
.
See the attached text file for more examples.

Scott R. Shannon, Table for n = 1..250.
Scott R. Shannon, Image for n = 20.

Cf. A092098 (regions), vertices (A331782), edges (A332376), A335413.

Sum of row(n) = A092098(n).

cp's OEIS Frontend

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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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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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.

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A367119 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 edges in the resulting planar graph.

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A335411 a(n) is the number of vertices formed by n-secting the angles of an equilateral triangle.

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A332377 One-third of 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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A354544 Table read by antidiagonals: T(n,k) (n >= 3, k >= 1) is the number of vertices formed in a regular n-gon by straight line segments when connecting the n corner vertices to the points dividing the sides into k equal parts.

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