A092867 -id:A092867 - cp's OEIS frontend

A332421 The number of regions inside a nonagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

154, 2754, 16858, 55098, 142318, 298350, 568162, 975294, 1585666, 2426292, 3588508, 5093604, 7067422, 9523746, 12612214, 16351218, 20924029, 26326026, 32789107, 40289238, 49093282, 59181228, 70852528

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

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

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

Cf. A332427 (n-gons), A332428 (vertices), A332429 (edges), A007678, A092867, A331452, A331929.

a(6)-a(23) from Lars Blomberg, May 16 2020

A333026 The number of vertices 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, 6, 16, 45, 111, 230, 448, 769, 1229, 1858, 2860, 4007, 5737, 7724, 10115, 13074, 17172, 21454, 27288, 33332, 40413, 48944, 59594, 70213, 82983, 97608, 113672, 131032, 152986, 174088, 201090, 228295, 258467, 292726, 328080, 365633, 412291, 460834, 512016

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), A333027 (edges), A007678, A092867, A331452, A331911, A332357, A332358.

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

A333139 The number of regions inside a decagon 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

220, 4220, 25220, 84280, 217800, 456640, 873090, 1501520, 2436020, 3736540, 5523970, 7830800, 10879460, 14665340, 19398660, 25173960, 32203320, 40502280, 50458120, 61995140, 75517160

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

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

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

Cf. A332417 (n-gons), A332418 (vertices), A332419 (edges), A007678, A092867, A331452, A331929.

a(6)-a(21) from Lars Blomberg, May 18 2020

A340639 The number of regions inside a Reuleaux triangle 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

1, 24, 145, 516, 1432, 3084, 6106, 10638, 17764, 27336, 41233, 58902, 82675, 111864, 149497, 194430, 250534, 316020, 395728, 487122, 596434, 720162, 865321, 1027974, 1216291, 1425348, 1664539, 1928022, 2226658, 2553204, 2920378, 3319536, 3764848, 4246638, 4780489, 5355414, 5988973

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 14 2021

nonn

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

Scott R. Shannon, Regions for n = 2.
Scott R. Shannon, Regions for n = 3.
Scott R. Shannon, Regions for n = 4.
Scott R. Shannon, Regions for n = 5.
Scott R. Shannon, Regions for n = 6.
Scott R. Shannon, Regions for n = 10.
Scott R. Shannon, Regions for n = 11.
Scott R. Shannon, Regions for n = 9 with random distance-based coloring.
Scott R. Shannon, Regions for n = 10 with random distance-based coloring.
Wikipedia, Reuleaux triangle.

Cf. A340644 (vertices), A340613 (edges), A340614 (n-gons), A007678, A092867.

A340644 The number of vertices on a Reuleaux triangle 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

3, 19, 120, 442, 1332, 2863, 5871, 10171, 17358, 26518, 40590, 57757, 81735, 110209, 148158, 192184, 248772, 313105, 393429, 483283, 593490, 715528, 861660, 1022281, 1211811, 1418515, 1659108, 1919842, 2220204, 2543527, 2912751, 3308305, 3755922, 4233730, 4770150, 5340529, 5977071

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 14 2021

nonn

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

Scott R. Shannon, Vertices for n = 2.
Scott R. Shannon, Vertices for n = 3.
Scott R. Shannon, Vertices for n = 4.
Scott R. Shannon, Vertices for n = 5.
Scott R. Shannon, Vertices for n = 6.
Scott R. Shannon, Vertices for n = 10.
Scott R. Shannon, Vertices for n = 11.
Wikipedia, Reuleaux triangle.

Cf. A340639 (regions), A340613 (edges), A340614 (n-gons), A007678, A092867.

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.

Original entry on oeis.org

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.

A328526 Number of regions in an equilateral triangle "frame" of size n.

Original entry on oeis.org

1, 12, 75, 249, 543, 1023, 1746, 2814, 4293, 6267, 8868, 12228, 16464, 21774, 28176, 35832, 45066, 56040, 68931, 84033, 101307, 120987, 143574, 169290, 198222, 230790, 267117, 307455, 352437, 402255, 457182, 517986, 584454, 656874, 735708, 821076, 913860

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 24 2020

nonn

A equilateral triangular "frame" of size n is formed from a triangular grid consisting of an outer edge of (n+1) points with the central grid of (n-5)*(n-6)/2 points removed. If n is less than 4 then no points or triangles are removed, and a(n) = A092867(n). From now on we assume n >= 4.
If we focus on the triangles rather than the points, the frame consists of a grid of equilateral triangles with the central block of (n-3)^2 triangles removed.
The resulting structure has an outer perimeter with 3*n points and an inner perimeter with 3*n-9 points, for a total of 6*n-9 perimeter points. The frame itself is the strip equilateral triangles pointing in alternate directions between the inner and outer perimeters such that the frame thickness equals the height of one triangle.
Now join every pair of perimeter points, both inner and outer, by a line segment, provided the line remains inside the frame. The sequence gives the number of regions in the resulting figure.
Like the square frame of A331776 only regions with 3 or 4 edges are formed.

Lars Blomberg, Table of n, a(n) for n = 1..65
Scott R. Shannon, Illustration for n=7.
Scott R. Shannon, Illustration for n=4 with random distance-based coloring.
Scott R. Shannon, Illustration for n=7 with random distance-based coloring.
Scott R. Shannon, Illustration for n=11 with random distance-based coloring.

Cf. A333030 (edges), A333031 (vertices), A333032 (3-gons), A333033 (4-gons), A331776 (square frame), A092867 (filled triangle).

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

A329714 Irregular table read by rows: Take a heptagon with all diagonals drawn, as in A329713. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Original entry on oeis.org

35, 7, 7, 0, 1, 504, 224, 112, 28, 2331, 1883, 1008, 273, 92, 7, 7658, 6314, 3416, 798, 182, 28, 18662, 17514, 8463, 2898, 714, 175, 28, 7, 0, 0, 0, 1, 40404, 35462, 18508, 5796, 1330, 266, 28

Offset: 1

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

See the links in A329713 for images of the heptagons.

			A heptagon with no other points along its edges, n = 1, contains 35 triangles, 7 quadrilaterals, 7 pentagons, 1 heptagon and no other n-gons, so the first row is [35,7,7,0,1]. A heptagon with 1 point dividing its edges, n = 2, contains 504 triangles, 224 quadrilaterals, 112 pentagons, 28 hexagons and no other n-gons, so the second row is [504,224,112,28].
Triangle begins:
35, 7, 7, 0, 1;
504, 224, 112, 28;
2331, 1883, 1008, 273, 92, 7;
7658, 6314, 3416, 798, 182, 28;
18662, 17514, 8463, 2898, 714, 175, 28, 7, 0, 0, 0, 1;
40404, 35462, 18508, 5796, 1330, 266, 28;
73248, 71596, 35777, 11669, 2654, 651, 70, 49;
The row sums are A329713.

Lars Blomberg, Table of n, a(n) for n = 1..251 (the first 27 rows)

Cf. A329713 (regions), A333112 (edges), A333113 (vertices), A331906, A007678, A092867, A331452.

cp's OEIS Frontend

A332421 The number of regions inside a nonagon 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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A333026 The number of vertices 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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A333139 The number of regions inside a decagon 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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A340639 The number of regions inside a Reuleaux triangle 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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A340644 The number of vertices on a Reuleaux triangle 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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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.

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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.

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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.

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A328526 Number of regions in an equilateral triangle "frame" of size n.

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A329714 Irregular table read by rows: Take a heptagon with all diagonals drawn, as in A329713. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

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