A344657 -id:A344657 - cp's OEIS frontend

A344279 Number of polygons formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.

1, 12, 102, 396, 1198, 2748, 5539, 10272, 16986, 26934, 41179, 60804, 84769, 119022, 157947, 206352, 268030, 347430, 432820, 543210, 659238, 801804, 970429, 1171662, 1371040, 1627398, 1904550, 2213712, 2555320, 2971260, 3373399, 3881838, 4399329, 4988502, 5610543, 6315312

Offset: 1

Scott R. Shannon, Jun 22 2021

nonn

Scott R. Shannon, Image for n = 2. In this and other images the triangle's points are highlighted as white dots while the outer open regions, which are not counted, are darkened. The key for the edge-number coloring is shown at the top-left of the image.
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 = 6.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.

Cf. A344657 (number of vertices), A344896 (number of edges), A346446 (number of k-gons), A092867 (number polygons inside the triangle), A343755 (number of regions), A345459, A344857.

a(n) = A344896(n) - A344657(n) + 1.

A344896 Number of polygon edges 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, 21, 186, 705, 2196, 5046, 10212, 19149, 31524, 50049, 77100, 114633, 159453, 225978, 298833, 390069, 508137, 663426, 824868, 1040727, 1258833, 1532565, 1859331, 2254938, 2628309, 3130227, 3664920, 4261503, 4917939, 5742696, 6503331, 7511280, 8507322, 9658806, 10855989, 12237066

Offset: 1

Scott R. Shannon, Jun 24 2021

nonn

See A344279 for images of the polygons.

Cf. A344279 (number of polygons), A344657 (number of vertices), A346446 (number of k-gons), A092867 (number polygons inside the triangle), A344899, A345650.

a(n) = A344279(n) + A344657(n) - 1.

A346446 Irregular triangle read by rows: T(n,k) = number of k-sided polygons formed when connecting infinite lines between all vertices and all points that divide the sides of an equilateral triangle into n equal parts, for k = 3, 4, ..., max_k.

Original entry on oeis.org

1, 12, 75, 24, 3, 258, 132, 6, 621, 525, 33, 19, 1308, 1272, 144, 24, 2505, 2628, 345, 61, 4434, 4734, 984, 102, 12, 6, 7365, 7992, 1347, 243, 30, 9, 11556, 12552, 2412, 366, 48, 17073, 19266, 3969, 804, 60, 3, 0, 3, 0, 1, 24786, 27672, 6954, 1206, 186, 34611, 39066, 9099, 1768, 198, 27

Offset: 1

Scott R. Shannon, Jul 18 2021

See A344279 for other images of the polygons.

			Connecting infinite lines between an equilateral triangle's three vertices and the two points along each side that divide the sides into three equal parts forms seventy-five triangles, twenty-four quadrilaterals and three pentagons, so row 3 is [75,24,3]. See the linked image.
The table begins:
       1;
      12;
      75,     24,      3;
     258,    132,      6;
     621,    525,     33,    19;
    1308,   1272,    144,    24;
    2505,   2628,    345,    61;
    4434,   4734,    984,   102,   12,   6;
    7365,   7992,   1347,   243,   30,   9;
   11556,  12552,   2412,   366,   48;
   17073,  19266,   3969,   804,   60,   3,  0,  3, 0, 1;
   24786,  27672,   6954,  1206,  186;
   34611,  39066,   9099,  1768,  198,  27;
   47028,  53688,  15318,  2676,  288,  24;
   63039,  72210,  18513,  3708,  396,  75,  0,  6;
   82746,  93570,  24930,  4536,  498,  54, 18;
  106536, 121080,  32988,  6622,  678, 117,  6,  3;
  134520, 155748,  46326,  9456, 1266, 102, 12;
  167895, 196179,  55527, 11410, 1638, 156, 12,  3;
  207294, 243294,  74796, 15396, 2106, 276, 42,  6;
  254034, 297069,  87648, 17715, 2388, 363, 18,  3;
  308022, 360228, 108264, 21858, 3090, 282, 42, 18;
  370818, 433902, 132651, 28210, 4311, 486, 42,  9;
  440952, 520044, 168156, 36228, 5484, 720, 78;
  521031, 614526, 189297, 39541, 5790, 780, 60, 15;
  612990, 723228, 232980, 49278, 8004, 822, 96;

Scott R. Shannon, Image of the k-gons for n=3.

Cf. A344279 (number of polygons), A344657 (number of vertices), A344896 (number of edges), A343755 (number of regions), A092867 (number polygons inside the triangle).

Sum of row(n) = A344279(n) = A344896(n) - A344657(n) + 1.

A357007 Number of vertices 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

3, 6, 15, 30, 51, 66, 111, 150, 171, 246, 303, 312, 435, 510, 543, 678, 771, 765, 975, 1059, 1131, 1326, 1455, 1488, 1731, 1878, 1899, 2178, 2355, 2376, 2703, 2886, 2955, 3270, 3444, 3420, 3891, 4110, 4191, 4485, 4803, 4878, 5295, 5526, 5544, 6078, 6351, 6396, 6915, 7206, 7311, 7794, 8115

Offset: 0

Scott R. Shannon, Sep 08 2022

nonn

See A356984 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.

Cf. A356984 (regions), A357008 (edges), A092866, A091908, A333026, A344657.

a(n) = A357008(n) - A356984(n) + 1 by Euler's formula.

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

A343755 Number of regions 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

7, 30, 144, 474, 1324, 2934, 5797, 10614, 17424, 27480, 41845, 61602, 85711, 120120, 159213, 207798, 269668, 349272, 434878, 545496, 661764, 804582, 973471, 1174980, 1374646, 1631304, 1908768, 2218254, 2560198, 2976486, 3378985, 3887796, 4405671, 4995240, 5617689, 6322878

Offset: 1

Scott R. Shannon, Jun 28 2021

nonn

The count of regions includes both the closed bounded polygons and the open unbounded areas surrounding these polygons with two edges that go to infinity. The number of unbounded areas appears to equal 6*(n^2 - n + 1).

See A344279 for further examples and images of the regions.

			a(1) = 7 as the three connected vertices of a triangle form one polygon along with six outer unbounded areas, seven regions in total.
a(2) = 30 as when the three vertices and three edges points are connected they form twelve polygons, all inside the triangle, along with eighteen outer unbounded areas, thirty regions in total.
a(2) = 144 as when the three vertices and six edges points are connected they form one hundred two polygons, seventy-five inside the triangle and twenty-seven outside, along with forty-two outer unbounded areas, one hundred forty-four regions in total.

Scott R. Shannon, Image for n = 1. In this and other images the triangle's vertices are highlighted as white dots while the outer open regions are cross-hatched. The key for the edge-number coloring is shown at the top-left of the image. Note the edge count for open areas also includes the two infinite edges
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 = 6.

Cf. A344279 (number of polygons), A344657 (number of vertices), A344896 (number of edges), A346446 (number of k-gons), A092867 (number polygons inside the triangle), A121205, A345025.

Conjectured formula: a(n) = A344279(n) + 6*(n^2 - n + 1).

Conjectured formula: a(n) = A344279(n) + A121205(n-1), for n>=7.

A386559 Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join an infinite straight line between every pair of points: a(n) is the number of points where lines intersect in the resulting graph.

Original entry on oeis.org

5, 65, 381, 1213, 3033, 6105, 12285, 20789, 33705, 51065, 79797, 110817, 161549, 216985, 284269, 367925, 489953, 609225, 785045, 952877, 1157749, 1404473

Offset: 1

Scott R. Shannon, Jul 26 2025

It appears that for n >= 3 the intersection that is furthest from the origin is formed by the crossing of the lines y = n/(n-1)*x + n and y = (n-1)/(n-2)*x - (n-1), along with the seven other symmetrically equivalent intersections. These intersections have a distance from the origin of approximately sqrt(8)*n^3 as n -> infinity.

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.

Cf. A386560 (regions), A386561 (edges), A386562 (k-gons), A146212, A347750, A344657, A345649.

a(n) = A386561(n) - A386560(n) + 1 by Euler's formula.

cp's OEIS Frontend

A344279 Number of polygons formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.

Views

Author

Keywords

Links

Crossrefs

Formula

A344896 Number of polygon edges formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.

Views

Author

Keywords

Comments

Crossrefs

Formula

A346446 Irregular triangle read by rows: T(n,k) = number of k-sided polygons formed when connecting infinite lines between all vertices and all points that divide the sides of an equilateral triangle into n equal parts, for k = 3, 4, ..., max_k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A357007 Number of vertices in an equilateral triangle when n internal equilateral triangles are drawn between the 3n points that divide each side into n+1 equal parts.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A343755 Number of regions formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A386559 Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join an infinite straight line between every pair of points: a(n) is the number of points where lines intersect in the resulting graph.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula