A343755 -id:A343755 - cp's OEIS frontend

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.

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

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.

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.

A364352 a(n) is the number of regions into which the plane is divided by n lines parallel to each edge of an equilateral triangle with side n such that the lines extend the parallel edge and divide the other edges into unit segments.

Original entry on oeis.org

7, 16, 30, 49, 73, 102, 136, 175, 219, 268, 322, 381, 445, 514, 588, 667, 751, 840, 934, 1033, 1137, 1246, 1360, 1479, 1603, 1732, 1866, 2005, 2149, 2298, 2452, 2611, 2775, 2944, 3118, 3297, 3481, 3670, 3864, 4063, 4267, 4476, 4690, 4909, 5133, 5362, 5596, 5835, 6079, 6328

Offset: 1

Nicolay Avilov, Jul 20 2023

Detailed instructions for drawing the lines. Along the edges of an equilateral triangle with side n, points are marked that divide the edges into unit segments. Draw all infinite straight lines that connect those points and are parallel to the edges of the triangle. For n = 1..5, the link shows the construction of these lines.

			a(1) = 1 + 3 + 3 = 7;
a(2) = 2^2 + 3*3 + 3 = 16;
a(5) = 5^2 + 3*9 + 3*6 + 3 = 73.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Nicolay Avilov, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A134238, A147875, A177862, A343755, A364401.

LinearRecurrence[{3,-3,1},{7,16,30},100] (* Paolo Xausa, Oct 16 2023 *)

a(n) = n*(5*n + 3)/2 + 3;
a(n) = A147875(n) + 3 = A134238(n+1) + 2.
From Stefano Spezia, Nov 23 2023: (Start)
O.g.f.: x*(7 - 5*x + 3*x^2)/(1 - x)^3.
E.g.f.: exp(x)*(3 + 4*x + 5*x^2/2) - 3. (End)

Edited by Peter Munn, Sep 02 2023

cp's OEIS Frontend

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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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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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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A364352 a(n) is the number of regions into which the plane is divided by n lines parallel to each edge of an equilateral triangle with side n such that the lines extend the parallel edge and divide the other edges into unit segments.

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