A092098 -id:A092098 - cp's OEIS frontend

A356984 Number of regions in an equilateral triangle when n internal equilateral triangles are drawn between the 3n points that divide each side into n+1 equal parts.

1, 4, 13, 28, 49, 70, 109, 148, 181, 244, 301, 334, 433, 508, 565, 676, 769, 811, 973, 1069, 1165, 1324, 1453, 1534, 1729, 1876, 1957, 2182, 2353, 2446, 2701, 2884, 3013, 3268, 3454, 3538, 3889, 4108, 4261, 4519, 4801, 4960, 5293, 5536, 5668, 6076, 6349, 6502, 6913, 7204, 7405, 7798, 8113

Offset: 0

Scott R. Shannon, Sep 08 2022

nonn

See A357007 for further images.

Scott R. Shannon, Table of n, a(n) for n = 0..250
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 = 5. This is the first term that forms intersections with non-simple vertices.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 50.
Scott R. Shannon, Image for n = 100.
Scott R. Shannon, Image for n = 200.
Talmon Silver, Classification of the intersection points and the number of regions

Cf. A357007 (vertices), A357008 (edges), A092867, A092098, A332953, A343755.

a(n) = A357008(n) - A357007(n) + 1 by Euler's formula.
Conjecture: a(n) = 3*n^2 + 1 for equilateral triangles that only contain simple vertices when cut by n internal equilateral triangles. This is never the case if (n + 1) mod 3 = 0 for n > 3.
a(n) = 1 + 3*n + T2(n) + 2*T3(n) + 3*T4(n); a(n) = 1 + 3*n^2 - T3(n) - 3*T4(n), where T2 is the number of internal vertices meeting exactly two segments (these vertices are labeled in the A357007 links as "4 ngons"), T3 is the number of internal vertices meeting exactly three segments ("6 ngons"), and T4 is the number of internal vertices meeting exactly four segments ("8 ngons"). - Talmon Silver, Sep 23 2022

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.

A277402 "3-Portolan numbers": number of regions formed by n-secting the angles of an equilateral triangle.

Original entry on oeis.org

1, 6, 19, 30, 61, 78, 127, 150, 217, 234, 331, 366, 469, 510, 631, 678, 817, 870, 1027, 1074, 1261, 1326, 1519, 1590, 1801, 1878, 2107, 2190, 2437, 2514, 2791, 2886, 3169, 3270, 3571, 3678, 3997, 4110, 4447, 4554, 4921, 5046, 5419, 5550, 5941, 6078, 6487, 6630, 7057, 7194

Offset: 1

Ethan Beihl, Oct 13 2016

nonn

I like the name "portolan numbers": cf. the rhumbline designs on medieval maps, constructed in a similar way.
The regions can be counted using an adaptation of Smiley and Wick's method in A092098: count regions assuming there are no redundant intersections, then subtract the number of regions that Ceva's Theorem says must vanish.
Off-diagonal redundant intersections occur for triples of integers 1 <= i, j, k < floor(n/2)-1 such that M(i)*M(j) = M(k), where M(x) is the ratio (sin(Pi(n-x)/(3n)))/(sin(Pi*x/(3n))). In the case 10|n, this corresponds to the charming identity (sin(7*Pi/30)*sin(8*Pi/30))/(sin(3*Pi/30)*sin(2*Pi/30)) = sin(9*Pi/30)/sin(1*Pi/30).
Differs from A092098 (which counts regions when *sides*, not angles, are n-sected) for the first time at the tenth term.
The above equation has solutions if and only if 10|n. This can be shown by rewriting the equation in exponential form, and using facts about vanishing sums of roots of unity to narrow the possibilities for n. (See Conway and Jones, 1976.) This is computationally feasible because A273096(6) = 1. - Ethan Beihl, Nov 26 2016

			For n=3, a(n) gives the 19 regions formed by the intersection of 3*2 lines: 3 pentagons, 3 quadrilaterals, 12 triangles, and 1 big central hexagon.

Lars Blomberg, Table of n, a(n) for n = 1..500
Lars Blomberg, Coloured illustration for n=3
Lars Blomberg, Coloured illustration for n=4
Lars Blomberg, Coloured illustration for n=19
Lars Blomberg, Coloured illustration for n=20
J. H. Conway and A. J. Jones, Trigonometric diophantine equations (On vanishing sums of roots of unity), Acta Arithmetica 30(3), 229-240 (1976).
Wikipedia, Rhumbline network

Cf. A092098, A335411 (vertices), A335412 (edges), A335413 (ngons).

regions[n_]:=
If[Mod[n,2]==0, 3n^2-6n+6, 3n^2-3n+1]-
  6*Length@
    Select[
     Flatten@
      With[
       {b=N@
          Table[
             1/2 - (Sqrt[3]/2)Tan[(60Degree/n)(n/2-i)],
             {i, 1, Floor[n/2]- 1}
             ]},
       Table[
        Abs[(1-b[[k]])b[[l]]b[[j]] - b[[k]](1-b[[l]])(1-b[[j]])],
        {j, 1, Floor[n/2]-1},
        {k, 1, Floor[n/2]-1},
        {l, 1, Floor[n/2]-1}]
       ],
     Chop@#==0&]

Empirical g.f.: x*(1 + 5*x + 12*x^2 + 6*x^3 + 18*x^4 + 6*x^5 + 18*x^6 + 6*x^7 + 18*x^8 - 6*x^9 + 29*x^10 + 13*x^11 - 6*x^12) / ((1-x)^3*(1+x)^2*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Oct 14 2016
Empirically for 12 < n <= 500: a(n) = a(n-2) + a(n-10) - a(n-12) + 120. - Lars Blomberg, Jun 08 2020
Empirical: a(2*k + 1) = 6*k*(2*k + 1) + 1, for k >= 0. - Ivan N. Ianakiev, Jun 27 2020
Empirical: 10*a(n) = 30*n^2 -45*n +23 +13*(-1)^n -15*(-1)^n*n - 24*b(n) where b(n) is the 10-periodic sequence 4, 0, -1, 0, -1, 0, -1, 0, -1, 0, 4, 0 .... of offset 0. - R. J. Mathar, Jul 05 2020

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.

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.

A345648 Number of polygons formed outside an n X n square when connecting all 4n points on the perimeter of the square by infinite lines.

Original entry on oeis.org

0, 0, 24, 228, 904, 2788, 5880, 13008, 22120, 37976, 58584, 95472, 125016, 195816, 255064, 337916, 444760, 611760, 719800, 978388, 1127088, 1411756, 1736776, 2174584, 2389552, 2995336, 3504768

Offset: 0

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

For n = 0 to n = 11 the number of polygons formed outside the n x n square is less than the number formed inside the square. This changes for n >= 12 when the number formed outside becomes greater. The ratio of the number of polygons outside to the number inside for n = 26 is about 1.07 . If this ratio is unbounded or approaches some finite value as n -> infinity is not known.

See A345459 for images of the polygons.

Cf. A345459 (number inside and outside with square), A255011 (number inside the square), A344993, A344857, A092098, A007678.

a(n) = A345459(n) - A255011(n).

A356044 Table read by antidiagonals: T(n,k) (n >= 3, k >= 1) is the number of regions 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

1, 6, 4, 19, 32, 11, 30, 96, 90, 24, 61, 188, 316, 246, 50, 78, 332, 580, 672, 462, 80, 127, 460, 1081, 1476, 1590, 856, 154, 150, 712, 1510, 2442, 2982, 2688, 1476, 220, 217, 916, 2306, 3528, 5370, 5504, 4861, 2420, 375, 246, 1204, 2930, 5310, 7742, 9440, 9288, 7360, 3630, 444

Offset: 3

Scott R. Shannon, Jul 31 2022

			The table begins:
1,    6,     19,     30,     61,     78,     127,    150,     217,     246,...
4,    32,    96,     188,    332,    460,    712,    916,     1204,    1488,...
11,   90,    316,    580,    1081,   1510,   2306,   2930,    3961,    4780,...
24,   246,   672,    1476,   2442,   3528,   5310,   7176,    9072,    11352,...
50,   462,   1590,   2982,   5370,   7742,   11390,  14742,   19650,   23982,...
80,   856,   2688,   5504,   9440,   13968,  20216,  27008,   34888,   43792,...
154,  1476,  4861,   9288,   16408,  23886,  34741,  45324,   59860,   73548,...
220,  2420,  7360,   14980,  25480,  37690,  54180,  72280,   93400,   116970,...
375,  3630,  11661,  22374,  39051,  57222,  82545,  108174,  142143,  175230,...
444,  5400,  15864,  32532,  55500,  82032,  118056, 157236,  203148,  254184,...
781,  7566,  23804,  45890,  79431,  116818, 167662, 220350,  288497,  356486,...
952,  10668, 31682,  63700,  107184, 160874, 226996, 303324,  390950,  489860,...
1456, 14070, 43546,  84270,  144976, 213720, 305656, 402510,  525586,  650550,...
1696, 18832, 55792,  111840, 187776, 282000, 397072, 530432,  683616,  855920,...
2500, 24072, 73645,  142732, 244410, 361012, 514795, 678912,  884800,  1096432,...
2466, 30078, 91080,  181746, 306522, 458334, 647766, 864234,  1114668, 1394586,...
4029, 38646, 117079, 227278, 387753, 573534, 816051, 1077414, 1401973, 1738918,...
.
.
See the attached text file for more examples and the cross references for further images.

Scott R. Shannon, Image of T(5,9) = 3961.
Scott R. Shannon, Image of T(6,8) = 7176.
Scott R. Shannon, Image of T(7,5) = 5370.
Scott R. Shannon, Image of T(8,4) = 5504.
Scott R. Shannon, Image of T(12,4) = 32532.
Scott R. Shannon, Image of T(14,6) = 160874.
Scott R. Shannon, Table for n = 3..25.

Cf. A007678 (first column), A092098 (first row), A108914 (second row).

T(n,1) = A007678(n).
T(3,k) = A092098(k).
T(4,k) = A108914(k).

A091910 Number of distinct distances between the intersection points in A091908 measured from the center of the equilateral triangle.

Original entry on oeis.org

1, 3, 4, 10, 11, 21, 20, 36, 36, 55, 56, 78, 79, 103, 103, 136, 135

Offset: 2

Hugo Pfoertner, Feb 19 2004

			a(2)=1: The 3 line segments intersect each other at the triangle center (r=0).
a(3)=3: There are 3 intersection points at r=0.2, 3 at r=0.25 and 6 at r=0.3779645, i.e. 3 different radii. See pictures given at link.

Hugo Pfoertner, Visualization of diagonal intersections in an equilateral triangle.
Hugo Pfoertner, Diagonal intersections in an equilateral triangle. Program and results.

Cf. A091908, A092098.

Fortran
```
! Program given at link.
```

A309360 Numbers n such that the number of interior intersection points A091908(n) of the n-intersected triangle decreases when the subdivision of the triangle is refined from n-1 to n cutting line segments.

Original entry on oeis.org

19, 23, 29, 39, 41, 47, 59, 65, 69, 71, 79, 83, 87, 89, 95, 103, 109, 111, 119, 125, 129, 131, 139, 143, 149, 151, 153, 155, 159, 167, 169, 179, 181, 191, 197, 199, 203, 207, 209, 215, 219, 223, 227, 229, 233, 237, 239, 251, 259, 263, 265, 269

Offset: 1

Hugo Pfoertner, Jul 26 2019

nonn

A091908(a(n)) < A091908(a(n)+1).

			a(1) = 19 because A091908(20)=961 < A091908(19)=972 is the first occurrence of a decrease in A091908.

Hugo Pfoertner, Illustration of first term, A091908(19)>A091908(20).

Cf. A091908, A092098, A309361.

cp's OEIS Frontend

A356984 Number of regions in an equilateral triangle when n internal equilateral triangles are drawn between the 3n points that divide each side into n+1 equal parts.

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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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A277402 "3-Portolan numbers": number of regions formed by n-secting the angles of an equilateral triangle.

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Mathematica

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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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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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A345648 Number of polygons formed outside an n X n square when connecting all 4n points on the perimeter of the square by infinite lines.

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A356044 Table read by antidiagonals: T(n,k) (n >= 3, k >= 1) is the number of regions 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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A091910 Number of distinct distances between the intersection points in A091908 measured from the center of the equilateral triangle.

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Fortran

A309360 Numbers n such that the number of interior intersection points A091908(n) of the n-intersected triangle decreases when the subdivision of the triangle is refined from n-1 to n cutting line segments.

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