A357007 -id:A357007 - 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

A357008 Number of edges 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, 9, 27, 57, 99, 135, 219, 297, 351, 489, 603, 645, 867, 1017, 1107, 1353, 1539, 1575, 1947, 2127, 2295, 2649, 2907, 3021, 3459, 3753, 3855, 4359, 4707, 4821, 5403, 5769, 5967, 6537, 6897, 6957, 7779, 8217, 8451, 9003, 9603, 9837, 10587, 11061, 11211, 12153, 12699, 12897, 13827, 14409, 14715

Offset: 0

Scott R. Shannon, Sep 08 2022

nonn

See A356984 and A357007 for images of the triangles.

Scott R. Shannon, Table of n, a(n) for n = 0..250

Cf. A356984 (regions), A357007 (vertices), A274586, A332376, A333027, A344896.

a(n) = A356984(n) + A357007(n) - 1 by Euler's formula.

Conjecture: a(n) = 6*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.

A357060 Number of vertices in a square when n internal squares are drawn between the 4n points that divide each side into n+1 equal parts.

Original entry on oeis.org

4, 8, 20, 40, 68, 88, 148, 168, 260, 296, 404, 436, 580, 632, 788, 840, 1028, 1072, 1300, 1384, 1604, 1688, 1940, 1972, 2308, 2408, 2708, 2808, 3140, 3220, 3604, 3696, 4084, 4232, 4628, 4716, 5188, 5336, 5764, 5908, 6404, 6496, 7060, 7224, 7732, 7928, 8468, 8524, 9220, 9368, 9988, 10216

Offset: 0

Scott R. Shannon, Sep 10 2022

nonn

The even values of n that yield squares with non-simple intersections are 32, 38, 44, 50, 54, 62, 76, 90, 98, ... .

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 squares with non-simple intersections.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 32. This is the first term with n mod 2 = 0 that forms squares with non-simple intersections.
Scott R. Shannon, Image for n = 200.

Cf. A357058 (regions), A357061 (edges), A355949, A355839, A355799, A357007 (triangle).

a(n) = A357061(n) - A357058(n) + 1 by Euler's formula.

Conjecture: a(n) = 4*n^2 + 4 for squares that only contain simple intersections when cut by n internal squares. This is never the case for odd n >= 5.

A357197 Number of vertices in a hexagon when n internal hexagons are drawn between the 6n points that divide each side into n+1 equal parts.

Original entry on oeis.org

6, 12, 30, 60, 102, 156, 222, 300, 390, 468, 606, 708, 870, 1020, 1152, 1356, 1542, 1740, 1950, 2112, 2406, 2652, 2910, 3072, 3462, 3756, 4062, 4350, 4710, 4974, 5406, 5772, 6126, 6540, 6918, 7260, 7782, 8220, 8646, 8946, 9606, 10032, 10590, 11052, 11568, 12156, 12702, 13116, 13830, 14388

Offset: 0

Scott R. Shannon, Sep 17 2022

nonn

Unlike similar dissections of the triangle and square, see A357007 and A357060, there is no obvious pattern for n values that yield hexagons with non-simple intersections; these n values begin 9, 11, 14, 19, 23, 27, 29, 32, 34, 35, 38, 39, 41, 43, ... .

Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 9. This is the first term that forms hexagons with non-simple intersections.
Scott R. Shannon, Image for n = 50.
Scott R. Shannon, Image for n = 150.

Cf. A357196 (regions), A357198 (edges), A330846, A357007 (triangle), A357060 (square).

a(n) = A357198(n) - A357196(n) + 1 by Euler's formula.

Conjecture: a(n) = 6*n^2 + 6 for hexagons that only contain simple intersections when cut by n internal hexagons.

A357235 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of vertices in an n-gon when k internal n-gons are drawn between the n*k points that divide each side into k+1 equal parts.

Original entry on oeis.org

3, 6, 4, 15, 8, 5, 30, 20, 10, 6, 51, 40, 25, 12, 7, 66, 68, 50, 30, 14, 8, 111, 88, 85, 60, 35, 16, 9, 150, 148, 130, 102, 70, 40, 18, 10, 171, 168, 185, 156, 119, 80, 45, 20, 11, 246, 260, 250, 222, 182, 136, 90, 50, 22, 12, 303, 296, 325, 300, 259, 208, 153, 100, 55, 24, 13

Offset: 3

Scott R. Shannon, Sep 19 2022

Conjecture: the only n-gons that contain non-simple intersections are the 3-gon (triangle), 4-gon (square), and 6-gon (hexagon).

			The table begins:
   3,  6, 15,  30,  51,  66, 111, 150, 171,  246,  303,  312,  435,  510,  543, ...
   4,  8, 20,  40,  68,  88, 148, 168, 260,  296,  404,  436,  580,  632,  788, ...
   5, 10, 25,  50,  85, 130, 185, 250, 325,  410,  505,  610,  725,  850,  985, ...
   6, 12, 30,  60, 102, 156, 222, 300, 390,  468,  606,  708,  870, 1020, 1152, ...
   7, 14, 35,  70, 119, 182, 259, 350, 455,  574,  707,  854, 1015, 1190, 1379, ...
   8, 16, 40,  80, 136, 208, 296, 400, 520,  656,  808,  976, 1160, 1360, 1576, ...
   9, 18, 45,  90, 153, 234, 333, 450, 585,  738,  909, 1098, 1305, 1530, 1773, ...
  10, 20, 50, 100, 170, 260, 370, 500, 650,  820, 1010, 1220, 1450, 1700, 1970, ...
  11, 22, 55, 110, 187, 286, 407, 550, 715,  902, 1111, 1342, 1595, 1870, 2167, ...
  12, 24, 60, 120, 204, 312, 444, 600, 780,  984, 1212, 1464, 1740, 2040, 2364, ...
  13, 26, 65, 130, 221, 338, 481, 650, 845, 1066, 1313, 1586, 1885, 2210, 2561, ...
  14, 28, 70, 140, 238, 364, 518, 700, 910, 1148, 1414, 1708, 2030, 2380, 2758, ...
  15, 30, 75, 150, 255, 390, 555, 750, 975, 1230, 1515, 1830, 2175, 2550, 2955, ...
See the attached text file for further examples.
See A357007, A357060, A357197 for more images of the n-gons.

Scott R. Shannon, Extended table for n = 3..50, k = 0..75.
Scott R. Shannon, Image of T(5,20) = 2005.
Scott R. Shannon, Image of T(7,10) = 707.

Cf. A357216 (regions), A357254 (edges), A357007 (triangle), A357060 (square), A357197 (hexagon), A007569, A146212.

T(n,k) = A357254(n,k) - A357216(n,k) + 1 by Euler's formula.
T(n,0) = n.
T(n,1) = 2n.
Conjectured formula for all columns for n >= 7: T(n,k) = n*k^2 + n.
T(3,k) = A357007(k).
T(4,k) = A357060(k).
T(6,k) = A357197(k).
Conjectured formula for all rows except for n = 3, 4, 6: T(n,k) = n*k^2 + n.

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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A357008 Number of edges 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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A357060 Number of vertices in a square when n internal squares are drawn between the 4n points that divide each side into n+1 equal parts.

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A357197 Number of vertices in a hexagon when n internal hexagons are drawn between the 6n points that divide each side into n+1 equal parts.

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A357235 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of vertices in an n-gon when k internal n-gons are drawn between the n*k points that divide each side into k+1 equal parts.

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