A274585 -id:A274585 - 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

A274586 Number of edges formed by sides and straight "chords" in a right triangle when each side is divided by vertices into n equal segments.

Original entry on oeis.org

3, 21, 132, 429, 1272, 2826, 5640, 10461, 17094, 26847, 41046, 61041, 84051, 118974, 157209, 204393, 264855, 346524, 428880, 541683, 654087, 793611, 961179, 1167468, 1357515, 1615209, 1891980, 2198019, 2530275, 2957808, 3341439, 3860652, 4371006, 4959636, 5572167, 6277722, 6950064, 7859406, 8763780, 9722571, 10687506, 11934912, 13029834, 14450598, 15805026, 17250795, 18863397, 20763204, 22372839, 24450474

Offset: 1

Cynthia Miaina Rasamimanananivo and Max Alekseyev, Jun 29 2016, Sep 02 2016

nonn

Cf. A092866, A092867, A274585.

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) = A274585(n) + A092867(n) - 1 = A092866(n) + A092867(n) + 3n - 1.

A092866 Number of intersections inside an equilateral triangular figure formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts. If three or more lines meet at an interior point this intersection is counted only once.

Original entry on oeis.org

0, 4, 49, 166, 543, 1237, 2511, 4762, 7777, 12262, 18933, 28504, 39078, 56065, 73879, 95962, 124653, 164761, 203259, 258646, 311233, 377932, 458793, 560755, 648936, 775258, 908893, 1056520, 1215087, 1428193, 1607871, 1866007, 2111488, 2399545, 2694010, 3040201, 3356433, 3811387, 4253074, 4720102, 5180466, 5806687, 6324906, 7035949, 7690900, 8392036, 9180330, 10136287, 10894551, 11930833

Offset: 1

Hugo Pfoertner, Mar 10 2004

nonn

A detailed example for n=5 is given at the Pfoertner link.

			a(2)=4 because there are 3 intersection points between the triangle medians and the line segments connecting the midpoints of the sides plus the intersection of the 3 medians at the centroid.

Jessica Gonzalez, Illustration of a(4)=166
Hugo Pfoertner, Intersections of diagonals in polygons of triangular shape.
Bjorn Poonen and Michael Rubinstein, The number of intersection points made by the diagonals of a regular polygon.
Cynthia Miaina Rasamimanananivo and Max Alekseyev, Sage program for this sequence
Index to OEIS, Sequences formed by drawing all diagonals in regular polygon

Cf. A092867 (regions formed by the diagonals), A274585 (points both inside and on the triangle sides), A274586 (edges).
Cf. A006561 (number of intersections of diagonals of regular n-gon), A091908 (intersections between line segments connecting vertices with subdivision points on opposite side).
If the boundary points are in general position, we get A367117, A213827, A367118, A367119. - N. J. A. Sloane, Nov 09 2023

Inter:= proc(p1x,p1y,p2x,p2y,q1x,q1y,q2x,q2y)
  local det,x,y;
  det:= p1x*q1y-p1x*q2y-p1y*q1x+p1y*q2x-p2x*q1y+p2x*q2y+p2y*q1x-p2y*q2x;
  if det = 0 then return NULL fi;
  x:= (p1x*p2y*q1x-p1x*p2y*q2x-p1x*q1x*q2y+p1x*q1y*q2x-p1y*p2x*q1x+p1y*p2x*q2x+p2x*q1x*q2y-p2x*q1y*q2x)/det;
  y:= (p1x*p2y*q1y-p1x*p2y*q2y-p1y*p2x*q1y+p1y*p2x*q2y-p1y*q1x*q2y+p1y*q1y*q2x+p2y*q1x*q2y-p2y*q1y*q2x)/det;
  if x >0 and y > 0 and x + y < 1 then [x,y]
  else NULL
  fi
end proc:
F:= proc(n) local A,B,C,Pairs,Pts;
     A:= [seq([j/n,0],j=0..n)];
     B:= [seq([0,j/n],j=0..n)];
     C:= [seq([j/n,1-j/n],j=0..n)];
     Pairs:= [seq(seq([A[i],B[j]],i=2..n+1),j=2..n+1),
              seq(seq([A[i],C[j]],i=1..n),j=1..n),
              seq(seq([B[i],C[j]],i=1..n),j=2..n+1)];
     Pts:= {seq(seq(Inter(op(Pairs[i][1]),op(Pairs[i][2]),op(Pairs[j][1]),op(Pairs[j][2])),j=1..i-1),i=2..nops(Pairs))};
     nops(Pts);
end proc:
map(F, [$1..20]); # Robert Israel, Jun 30 2016

Inter[{p1x_, p1y_}, {p2x_, p2y_}, {q1x_, q1y_}, {q2x_, q2y_}] := Module[ {det, x, y}, det = p1x q1y - p1x q2y - p1y q1x + p1y q2x - p2x q1y + p2x q2y + p2y q1x - p2y q2x; If[det == 0, Return[Nothing]]; x = (p1x p2y q1x - p1x p2y q2x - p1x q1x q2y + p1x q1y q2x - p1y p2x q1x + p1y p2x q2x + p2x q1x q2y - p2x q1y q2x)/det; y = (p1x p2y q1y - p1x p2y q2y - p1y p2x q1y + p1y p2x q2y - p1y q1x q2y + p1y q1y q2x + p2y q1x q2y - p2y q1y q2x)/det; If[x > 0 && y > 0 && x + y < 1, {x, y}, Nothing]];
F[n_] := F[n] = Module[{A, B, K, Pairs, Pts}, A = Table[{j/n, 0}, {j, 0, n}]; B = Table[{0, j/n}, {j, 0, n}]; K = Table[{j/n, 1 - j/n}, {j, 0, n}]; Pairs = {Table[Table[{A[[i]], B[[j]]}, {i, 2, n+1}], {j, 2, n+1}], Table[Table[{A[[i]], K[[j]]}, {i, 1, n}], {j, 1, n}], Table[Table[ {B[[i]], K[[j]]}, {i, 1, n}], {j, 2, n+1}]} // Flatten[#, 2]&; Pts = Table[Table[Inter[Pairs[[i, 1]], Pairs[[i, 2]], Pairs[[j, 1]], Pairs[[j, 2]]], {j, 1, i-1}], {i, 2, Length[Pairs]}]; Flatten[Pts, 1] // Union // Length];
Table[Print[n, " ", F[n]]; F[n], {n, 1, 20}] (* Jean-François Alcover, Apr 11 2019, after Robert Israel *)

a(n) = A274585(n) - 3n.

a(1) = 0 prepended by Max Alekseyev, Jun 29 2016
a(4) corrected and a(6)-a(20) added by Cynthia Miaina Rasamimanananivo, Jun 28 2016
a(20) corrected by Robert Israel, Jun 30 2016
a(21)-a(50) from Cynthia Miaina Rasamimanananivo, Jun 30 - Aug 23, 2016
"Equilateral" added to definition by N. J. A. Sloane, May 13 2020

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.

A333031 Number of vertices in an equilateral triangle "frame" of size n (see Comments in A328526 for definition).

Original entry on oeis.org

3, 10, 58, 183, 408, 777, 1323, 2142, 3276, 4773, 6717, 9264, 12507, 16554, 21351, 27090, 34047, 42318, 52008, 63453, 76566, 91371, 108249, 127608, 149487, 173982, 201072, 231225, 265002, 302487, 343857, 389856, 440175, 494670, 553611, 617610, 687477, 763320

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 05 2020

nonn

For n<=3 the terms equal A274585(n). See A328526 for images of the triangular frame.

Lars Blomberg, Table of n, a(n) for n = 1..65

Cf. A328526 (regions), A333030 (edges), A333032 (3-gons), A333033 (4-gons), A331776 (square frame), A274586 (filled triangle).

a(12) and beyond from Lars Blomberg, May 01 2020

A366483 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of vertices in the resulting planar graph.

Original entry on oeis.org

3, 6, 22, 108, 300, 919, 1626, 3558, 5824, 9843, 14352, 23845, 30951, 47196, 62773, 82488, 104544, 144784, 173694, 230008, 276388, 336927, 403452, 509218, 582417, 702228, 824956, 969387, 1098312, 1321978, 1463580, 1724190, 1952509, 2221497, 2505169, 2846908, 3103788, 3556143, 3978763, 4444003

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Nov 09 2023

nonn

We start with the three corner points of the triangle, and add n further points along each edge. Including the corner points, we end up with n+2 points along each edge, and the edge is divided into n+1 line segments.

Each of the n points added to an edge is joined by 2*n chords to the points that were added to the other two edges. There are 3*n^2 chords.

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. A366484 (interior vertices), A366485 (edges), A366486 (regions).
If the 3*n points are placed "in general position" instead of uniformly, we get sequences A366478, A365929, A366932, A367015.
If the 3*n points are placed uniformly and we also draw chords from the three corner points of the triangle to these 3*n points, we get A274585, A092866, A274586, A092867.

a(n) = A366485(n) - A366486(n) + 1 (Euler).

A366484 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of interior vertices in the resulting planar graph.

Original entry on oeis.org

0, 0, 13, 96, 285, 901, 1605, 3534, 5797, 9813, 14319, 23809, 30912, 47154, 62728, 82440, 104493, 144730, 173637, 229948, 276325, 336861, 403383, 509146, 582342, 702150, 824875, 969303, 1098225, 1321888, 1463487, 1724094, 1952410, 2221395, 2505064, 2846800, 3103677, 3556029, 3978646, 4443883

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Nov 09 2023

nonn

See A366483 for further information.

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. A366483 (vertices), A366485 (edges), A366486 (regions).
If the 3*n points are placed "in general position" instead of uniformly, we get sequences A366478, A365929, A366932, A367015.
If the 3*n points are placed uniformly and we also draw chords from the three corner points of the triangle to these 3*n points, we get A274585, A092866, A274586, A092867.

a(n) = A366485(n) - A366486(n) - 3*n - 2 (Euler).

A366486 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of regions in the resulting planar graph.

Original entry on oeis.org

1, 4, 27, 130, 385, 1044, 2005, 4060, 6831, 11272, 16819, 26436, 35737, 52147, 69984, 92080, 117952, 157770, 193465, 249219, 302670, 368506, 443026, 546462, 635125, 757978, 890133, 1041775, 1191442, 1407324, 1581058, 1837417, 2085096, 2365657, 2670429, 3018822, 3328351, 3771595, 4213602

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Nov 09 2023

nonn

See A366483 for further information.

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. A366483 (vertices), A366484 (interior vertices), A366485 (edges).
If the 3*n points are placed "in general position" instead of uniformly, we get sequences A366478, A365929, A366932, A367015.
If the 3*n points are placed uniformly and we also draw chords from the three corner points of the triangle to these 3*n points, we get A274585, A092866, A274586, A092867.

a(n) = A366485(n) - A366483(n) + 1 (Euler).

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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A274586 Number of edges formed by sides and straight "chords" in a right triangle when each side is divided by vertices into n equal segments.

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A092866 Number of intersections inside an equilateral triangular figure formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts. If three or more lines meet at an interior point this intersection is counted only once.

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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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A333031 Number of vertices in an equilateral triangle "frame" of size n (see Comments in A328526 for definition).

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A366483 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of vertices in the resulting planar graph.

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A366484 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of interior vertices in the resulting planar graph.

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A366486 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of regions in the resulting planar graph.

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