A367117 -id:A367117 - 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

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.

A334698 a(n) is the total number of points (both boundary and interior points) in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter), where the interior points are counted with multiplicity.

Original entry on oeis.org

5, 58, 375, 1376, 3685, 8130, 15743, 27760, 45621, 70970, 105655, 151728, 211445, 287266, 381855, 498080, 639013, 807930, 1008311, 1243840, 1518405, 1836098, 2201215, 2618256, 3091925, 3627130, 4228983, 4902800, 5654101, 6488610, 7412255, 8431168, 9551685, 10780346, 12123895, 13589280, 15183653, 16914370, 18788991

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 18 2020

An equivalent definition: Place n-1 points in general position on each side of a square, and join every pair of the 4*n+4 boundary points by a chord; sequence gives number of vertices in the resulting planar graph. "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 and N. J. A. Sloane, Nov 05 2023
Equivalently, this is A334697(n) + 4*n.
This is an upper bound on A331449.

Colin Barker, Table of n, a(n) for n = 1..1000
Scott R. Shannon, Colored illustration for n = 2
Scott R. Shannon, Illustration for n=3 showing interior vertices color-coded according to multiplicity.
Scott R. Shannon, "General position" image for n = 1.
Scott R. Shannon, "General position" image for n = 2.
Scott R. Shannon, "General position" image for n = 3.
Scott R. Shannon, "General position" image for n = 4.
Scott R. Shannon, "General position" image for n = 5.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A255011, A331449, A334690-A334698.

For the "general position" version, see also A367121 (regions), A367122 (edges), and A367117.

A334698[n_]:=n(17n^3-30n^2+19n+4)/2;Array[A334698,50] (* or *)
LinearRecurrence[{5,-10,10,-5,1},{5,58,375,1376,3685},50] (* Paolo Xausa, Nov 14 2023 *)

Vec(x*(5 + 33*x + 135*x^2 + 31*x^3) / (1 - x)^5 + O(x^40)) \\ Colin Barker, May 31 2020

Theorem: a(n) = n*(17*n^3-30*n^2+19*n+4)/2.
From Colin Barker, May 27 2020: (Start)
G.f.: x*(5 + 33*x + 135*x^2 + 31*x^3) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)
a(n+1) = A367122(n) - A367121(n) + 1 by Euler's formula.

Edited by N. J. A. Sloane, Nov 13 2023

A367183 Table read by antidiagonals: Place k points in general position on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives number of vertices in the resulting planar graph.

Original entry on oeis.org

3, 12, 5, 72, 58, 10, 282, 375, 185, 19, 795, 1376, 1155, 451, 42, 1818, 3685, 4090, 2734, 938, 57, 3612, 8130, 10700, 9478, 5523, 1711, 135, 6492, 15743, 23235, 24463, 18858, 9981, 2943, 171, 10827, 27760, 44485, 52639, 48230, 33771, 16740, 4646, 341

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Nov 08 2023

"In general position" implies that the internal lines (or chords) formed from the n*k edge points only have simple intersections; there is no interior points where three or more such chords meet. Note that for even-n n-gons, with n>=6, the chords from the n corner points do create non-simple intersections.

			The table begins:
3, 12, 72, 282, 795, 1818, 3612, 6492, 10827, 17040, 25608, 37062, 51987,...
5, 58, 375, 1376, 3685, 8130, 15743, 27760, 45621, 70970, 105655, 151728,...
10, 185, 1155, 4090, 10700, 23235, 44485, 77780, 126990, 196525, 291335,...
19, 451, 2734, 9478, 24463, 52639, 100126, 174214, 283363, 437203, 646534,...
42, 938, 5523, 18858, 48230, 103152, 195363, 338828, 549738, 846510, 1249787,...
57, 1711, 9981, 33771, 85849, 182847, 345261, 597451, 967641, 1487919, 2194237,...
135, 2943, 16740, 56106, 141885, 301185, 567378, 980100, 1585251, 2434995,...
171, 4646, 26336, 87831, 221351, 468746, 881496, 1520711, 2457131, 3771126,...
341, 7128, 39666, 131450, 330165, 697686, 1310078, 2257596, 3644685, 5589980,...
313, 10204, 57199, 189214, 474361, 1000948, 1877479, 3232654, 5215369, 7994716,...
728, 14677, 80457, 264602, 661570, 1393743, 2611427, 4492852, 7244172,...
771, 19909, 109586, 359892, 898591, 1891121, 3540594, 6087796, 9811187,...
1380, 27030, 146565, 479370, 1194600, 2511180, 4697805, 8072940, 13004820,...
1393, 35085, 191353, 625477, 1557297, 3271213, 6116185, 10505733,......
.
.
.

Scott R. Shannon, Image for T(5,3).
Scott R. Shannon, Image for T(6,2).
Scott R. Shannon, Image for T(7,1).
Scott R. Shannon, Image for T(8,1).

Cf. A367117 (first row), A334698 (second row), A007569 (first column), A366253 (regions), A367190 (edges).

T(n,k) = A367190(n,k) - A366253(n,k) + 1 by Euler's formula.
T(3,k) = A367117(k) = (9/4)*k^4 + 3*k^3 + (3/4)*k^2 + 3*k + 3.
Conjectured:
T(4,k) = A334698(k+1) = (17/2)*k^4 + 19*k^3 + (31/2)*k^2 + 10*k + 5.
T(5,k) = (45/2)*k^4 + 60*k^3 + 60*k^2 + (65/2)*k + 10.
T(6,k) = (195/4)*k^4 + (285/2)*k^3 + (627/4)*k^2 + 84*k + 19.
T(7,k) = (371/4)*k^4 + 287*k^3 + (1337/4)*k^2 + 182*k + 42.
T(8,k) = 161*k^4 + 518*k^3 + 627*k^2 + 348*k + 57.
T(9,k) = 261*k^4 + 864*k^3 + (2151/2)*k^2 + (1215/2)*k + 135.
T(10,k) = (1605/4)*k^4 + (2715/2)*k^3 + (6905/4)*k^2 + 990*k + 171.

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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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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A334698 a(n) is the total number of points (both boundary and interior points) in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter), where the interior points are counted with multiplicity.

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A367183 Table read by antidiagonals: Place k points in general position on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives number of vertices in the resulting planar graph.

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