A092867 -id:A092867 - cp's OEIS frontend

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.

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.

A332953 The number of regions formed inside an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

Original entry on oeis.org

1, 5, 18, 52, 125, 257, 486, 832, 1333, 2027, 3048, 4304, 6057, 8167, 10749, 13929, 18058, 22664, 28533, 34981, 42519, 51425, 62118, 73473, 86768, 101902, 118695, 137138, 159147, 181752, 208813, 237209, 268614, 303718, 340882, 380811, 427540, 477134, 530047

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 04 2020

nonn

The terms are from numeric computation - no formula for a(n) is currently known.
Equivalently, this is also the number of regions formed when all the integer points along the x and y axes with 0 <= x <= n and 0 <= y <= n are joined by straight line segments.
If instead one takes points on the x and y axes with coordinates 1, 1/2, 1/3, 1/4, ..., 1/n, 0, and joins them all by line segments, the resulting figure contains only triangles and quadrilaterals, and the number of regions is given by A332358 (and more generally by A332357 if there are m+1 such points on the x axis and n+1 such points on the y axis).

Lars Blomberg, Table of n, a(n) for n = 1..70
Scott R. Shannon, Illustration for n = 2.
Scott R. Shannon, Illustration for n = 3.
Scott R. Shannon, Illustration for n = 4.
Scott R. Shannon, Illustration for n = 5.
Scott R. Shannon, Illustration for n = 6.
Scott R. Shannon, Illustration for n = 8.
Scott R. Shannon, Illustration for n = 10.
Scott R. Shannon, Illustration for n = 12.
Scott R. Shannon, Illustration for n = 15.
Scott R. Shannon, Illustration for n = 5 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 10 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 15 with random distance-based coloring.

Cf. A333025 (n-gons), A333026 (vertices), A333027 (edges), A007678, A092867, A331452, A331911, A332357, A332358.

a(16) and beyond from Lars Blomberg, May 26 2020

A333025 Irregular table read by rows: Take an isosceles triangle with its equal length sides divided into n equal parts with all diagonals drawn, as in A332953. Then T(n,k) = number of k-sided polygons in that figure for k>=3.

Original entry on oeis.org

1, 5, 14, 3, 1, 29, 19, 4, 50, 66, 9, 81, 164, 12, 134, 313, 37, 2, 219, 546, 60, 7, 359, 853, 112, 9, 556, 1294, 160, 16, 1, 779, 1940, 283, 43, 3, 1105, 2780, 360, 53, 6, 1540, 3750, 670, 91, 5, 1, 2087, 5064, 873, 132, 11, 2806, 6625, 1144, 164, 7, 3

Offset: 1

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

See the links in A332953 for images of the triangles.

			Table begins:
1;
5;
14, 3, 1;
29, 19, 4;
50, 66, 9;
81, 164, 12;
134, 313, 37, 2;
219, 546, 60, 7;
359, 853, 112, 9;
556, 1294, 160, 16, 1;
779, 1940, 283, 43, 3;
1105, 2780, 360, 53, 6;
1540, 3750, 670, 91, 5, 1;
2087, 5064, 873, 132, 11;
2806, 6625, 1144, 164, 7, 3;
The row sums are A332953.

Lars Blomberg, Table of n, a(n) for n = 1..613 (the first 70 rows)

Cf. A332953 (regions), A333026 (vertices), A333027 (edges), A007678, A092867, A331452, A331911, A332357, A332358.

A333027 The number of edges formed on an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

Original entry on oeis.org

3, 10, 33, 96, 235, 486, 933, 1600, 2561, 3884, 5907, 8310, 11793, 15890, 20863, 27002, 35229, 44117, 55820, 68312, 82931, 100368, 121711, 143685, 169750, 199509, 232366, 268169, 312132, 355839, 409902, 465503, 527080, 596443, 668961, 746443, 839830, 937967

Offset: 1

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

nonn

See the links in A332953 for images of the triangles.

Lars Blomberg, Table of n, a(n) for n = 1..70

Cf. A332953 (regions), A333025 (n-gons), A333026 (vertices), A007678, A092867, A331452, A331911, A332357, A332358.

a(16) and beyond from Lars Blomberg, May 26 2020

A344657 Number of vertices formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.

Original entry on oeis.org

3, 10, 85, 310, 999, 2299, 4674, 8878, 14539, 23116, 35922, 53830, 74685, 106957, 140887, 183718, 240108, 315997, 392049, 497518, 599596, 730762, 888903, 1083277, 1257270, 1502830, 1760371, 2047792, 2362620, 2771437, 3129933, 3629443, 4107994, 4670305, 5245447, 5921755

Offset: 1

Scott R. Shannon, Jun 24 2021

nonn

See A344279 for other images of the polygons.

Scott R. Shannon, Image of the vertices for n = 3.
Scott R. Shannon, Image of the vertices for n = 4.

Cf. A344279 (number of polygons), A344896 (number of edges), A092867 (number polygons inside the triangle), A346446 (number of k-gons), A345649, A146212.

a(n) = A344896(n) - A344279(n) + 1.

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.

Original entry on oeis.org

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.

A329713 The number of regions inside a heptagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

50, 868, 5594, 18396, 48462, 101794, 195714, 336504, 549704, 841890, 1249676, 1774612, 2468572, 3328234, 4414054, 5725034, 7336855, 9233098, 11513419, 14149296, 17254434, 20805554, 24928380, 29573348, 34902155, 40861422, 47613161

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 07 2020

The terms are from numeric computation - no formula for a(n) is currently known.

Scott R. Shannon, Heptagon regions for n = 1.
Scott R. Shannon, Heptagon regions for n = 2.
Scott R. Shannon, Heptagon regions for n = 3.
Scott R. Shannon, Heptagon regions for n = 4.
Scott R. Shannon, Heptagon regions with random distance-based coloring for n = 1.
Scott R. Shannon, Heptagon regions with random distance-based coloring for n = 2.
Scott R. Shannon, Heptagon regions with random distance-based coloring for n = 3.
Wikipedia, Heptagon.

Cf. A329714 (n-gons), A333112 (edges), A333113 (vertices), A007678, A092867, A331452, A331931.

a(8)-a(27) from Lars Blomberg, May 13 2020

A331932 Triangle read by rows: Take a hexagon with all diagonals drawn, as in A331931. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+4.

Original entry on oeis.org

18, 6, 0, 264, 108, 36, 0, 1344, 654, 252, 12, 6, 4164, 2772, 1020, 228, 24, 0, 10038, 7758, 2424, 516, 72, 24, 0, 21108, 16188, 6060, 1128, 156, 0, 0, 0, 39690, 32022, 13368, 3654, 432, 48, 0, 0, 0, 68052, 56616, 22980, 6084, 888, 120, 12, 0, 0, 0

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 01 2020

See the links in A331931 for images of the hexagons.

			A hexagon with no other points along its edges, n = 1, contains 18 triangles, 6 quadrilaterals and no other n-gons, so the first row is [18,6,0]. A hexagon with 1 point dividing its edges, n = 2, contains 264 triangles, 108 quadrilaterals, 36 pentagons and no other n-gons, so the second row is [264,108,36,0].
Triangle begins:
  18,6,0
  264,108,36,0
  1344,654,252,12,6
  4164,2772,1020,228,24,0
  10038,7758,2424,516,72,24,0
  21108,16188,6060,1128,156,0,0,0
  39690,32022,13368,3654,432,48,0,0,0
  68052,56616,22980,6084,888,120,12,0,0,0
The row sums are A331931.

Lars Blomberg, Table of n, a(n) for n = 1..525 (the first 30 rows)
Wikipedia, Hexagon.

Cf. A331931 (regions), A330845 (edges), A330846 (vertices), A331906, A007678, A092867, A331452.

cp's OEIS Frontend

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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A332953 The number of regions formed inside an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

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A333025 Irregular table read by rows: Take an isosceles triangle with its equal length sides divided into n equal parts with all diagonals drawn, as in A332953. Then T(n,k) = number of k-sided polygons in that figure for k>=3.

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A333027 The number of edges formed on an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

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A344657 Number of vertices formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.

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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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A329713 The number of regions inside a heptagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

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A331932 Triangle read by rows: Take a hexagon with all diagonals drawn, as in A331931. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+4.

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