A357058 -id:A357058 - cp's OEIS frontend

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.

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.

A357196 Number of regions 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

1, 7, 25, 55, 97, 151, 217, 295, 385, 475, 601, 715, 865, 1015, 1159, 1351, 1537, 1735, 1945, 2131, 2401, 2647, 2905, 3115, 3457, 3751, 4057, 4357, 4705, 5005, 5401, 5767, 6133, 6535, 6925, 7303, 7777, 8215, 8653, 9025, 9601, 10051, 10585, 11071, 11587, 12151, 12697, 13171, 13825, 14395, 14989

Offset: 0

Scott R. Shannon, Sep 17 2022

nonn

Unlike similar dissections of the triangle and square, see A356984 and A357058, 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. A357197 (vertices), A357198 (edges), A331931, A356984 (triangle), A357058 (square).

Cf. A227776 (6*n^2 + 1).

a(n) = A357198(n) - A357197(n) + 1 by Euler's formula.

Conjecture: a(n) = 6*n^2 + 1 for hexagons that only contain simple intersections when cut by n internal hexagons.

A357061 Number of edges 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, 12, 36, 76, 132, 180, 292, 348, 516, 604, 804, 892, 1156, 1284, 1572, 1708, 2052, 2180, 2596, 2796, 3204, 3412, 3876, 4012, 4612, 4860, 5412, 5668, 6276, 6508, 7204, 7460, 8172, 8524, 9252, 9516, 10372, 10740, 11532, 11900, 12804, 13100, 14116, 14532, 15468, 15940, 16932, 17196, 18436

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, ... .

See A357058 and A357060 for images of the squares.

Cf. A357058 (regions), A357060 (vertices), A355948, A355840, A355800, A357008 (triangle).

a(n) = A357058(n) + A357060(n) - 1 by Euler's formula.

Conjecture: a(n) = 8*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.

A357216 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of regions 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

1, 4, 1, 13, 5, 1, 28, 17, 6, 1, 49, 37, 21, 7, 1, 70, 65, 46, 25, 8, 1, 109, 93, 81, 55, 29, 9, 1, 148, 145, 126, 97, 64, 33, 10, 1, 181, 181, 181, 151, 113, 73, 37, 11, 1, 244, 257, 246, 217, 176, 129, 82, 41, 12, 1, 301, 309, 321, 295, 253, 201, 145, 91, 45, 13, 1

Offset: 3

Scott R. Shannon, Sep 18 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:
  1,  4, 13,  28,  49,  70, 109, 148, 181,  244,  301,  334,  433,  508,  565, ...
  1,  5, 17,  37,  65,  93, 145, 181, 257,  309,  401,  457,  577,  653,  785, ...
  1,  6, 21,  46,  81, 126, 181, 246, 321,  406,  501,  606,  721,  846,  981, ...
  1,  7, 25,  55,  97, 151, 217, 295, 385,  475,  601,  715,  865, 1015, 1159, ...
  1,  8, 29,  64, 113, 176, 253, 344, 449,  568,  701,  848, 1009, 1184, 1373, ...
  1,  9, 33,  73, 129, 201, 289, 393, 513,  649,  801,  969, 1153, 1353, 1569, ...
  1, 10, 37,  82, 145, 226, 325, 442, 577,  730,  901, 1090, 1297, 1522, 1765, ...
  1, 11, 41,  91, 161, 251, 361, 491, 641,  811, 1001, 1211, 1441, 1691, 1961, ...
  1, 12, 45, 100, 177, 276, 397, 540, 705,  892, 1101, 1332, 1585, 1860, 2157, ...
  1, 13, 49, 109, 193, 301, 433, 589, 769,  973, 1201, 1453, 1729, 2029, 2353, ...
  1, 14, 53, 118, 209, 326, 469, 638, 833, 1054, 1301, 1574, 1873, 2198, 2549, ...
  1, 15, 57, 127, 225, 351, 505, 687, 897, 1135, 1401, 1695, 2017, 2367, 2745, ...
  1, 16, 61, 136, 241, 376, 541, 736, 961, 1216, 1501, 1816, 2161, 2536, 2941, ...
  ...
See the attached text file for further examples.
See A356984, A357058, A357196 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) = 2001.
Scott R. Shannon, Image of T(7,10) = 701.

Cf. A357235 (vertices), A357254 (edges), A356984 (triangle), A357058 (square), A357196 (hexagon), A007678, A344857.

T(n,k) = A357254(n,k) - A357235(n,k) + 1 by Euler's formula.
T(n,0) = 1.
T(n,1) = n + 1.
Conjectured formula for all columns for n >= 7: T(n,k) = n*k^2 + 1.
T(3,k) = A356984(k).
T(4,k) = A357058(k).
T(6,k) = A357196(k).
Conjectured formula for all rows except for n = 3, 4, 6: T(n,k) = n*k^2 + 1.

A357254 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of edges 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, 9, 4, 27, 12, 5, 57, 36, 15, 6, 99, 76, 45, 18, 7, 135, 132, 95, 54, 21, 8, 219, 180, 165, 114, 63, 24, 9, 297, 292, 255, 198, 133, 72, 27, 10, 351, 348, 365, 306, 231, 152, 81, 30, 11, 489, 516, 495, 438, 357, 264, 171, 90, 33, 12, 603, 604, 645, 594, 511, 408, 297, 190, 99, 36, 13

Offset: 3

Scott R. Shannon, Sep 20 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,  9,  27,  57,  99, 135,  219,  297,  351,  489,  603,  645,  867, 1017, ...
   4, 12,  36,  76, 132, 180,  292,  348,  516,  604,  804,  892, 1156, 1284, ...
   5, 15,  45,  95, 165, 255,  365,  495,  645,  815, 1005, 1215, 1445, 1695, ...
   6, 18,  54, 114, 198, 306,  438,  594,  774,  942, 1206, 1422, 1734, 2034, ...
   7, 21,  63, 133, 231, 357,  511,  693,  903, 1141, 1407, 1701, 2023, 2373, ...
   8, 24,  72, 152, 264, 408,  584,  792, 1032, 1304, 1608, 1944, 2312, 2712, ...
   9, 27,  81, 171, 297, 459,  657,  891, 1161, 1467, 1809, 2187, 2601, 3051, ...
  10, 30,  90, 190, 330, 510,  730,  990, 1290, 1630, 2010, 2430, 2890, 3390, ...
  11, 33,  99, 209, 363, 561,  803, 1089, 1419, 1793, 2211, 2673, 3179, 3729, ...
  12, 36, 108, 228, 396, 612,  876, 1188, 1548, 1956, 2412, 2916, 3468, 4068, ...
  13, 39, 117, 247, 429, 663,  949, 1287, 1677, 2119, 2613, 3159, 3757, 4407, ...
  14, 42, 126, 266, 462, 714, 1022, 1386, 1806, 2282, 2814, 3402, 4046, 4746, ...
  15, 45, 135, 285, 495, 765, 1095, 1485, 1935, 2445, 3015, 3645, 4335, 5085, ...
  ...
See the attached text file for further examples.
See A356984, A357058, A357196 for images of the n-gons.

Scott R. Shannon, Extended table for n = 3..50, k = 0..75.

Cf. A357216 (regions), A357235 (vertices), A357008 (triangle), A357061 (square), A357198 (hexagon), A356984, A357058, A357196, A135565, A344899.

T(n,k) = A357216(n,k) + A357235(n,k) - 1 by Euler's formula.
T(n,0) = n.
T(n,1) = 3n.
Conjectured formula for all columns for n >= 7: T(n,k) = 2n*k^2 + n.
T(3,k) = A357008(k).
T(4,k) = A357061(k).
T(6,k) = A357198(k).
Conjectured formula for all rows except for n = 3, 4, 6: T(n,k) = 2n*k^2 + n.

A358556 Triangle read by rows: T(n,k) is the number of regions formed when n points are placed along each edge of a square that divide the edges into n+1 equal parts and a line is continuously drawn from the current point to that k points, 2 <= k <= 2*n, counterclockwise around the square until the starting point is again reached.

Original entry on oeis.org

2, 5, 21, 2, 5, 5, 4, 61, 2, 5, 29, 5, 73, 25, 105, 2, 5, 25, 5, 5, 31, 141, 11, 157, 2, 5, 5, 5, 85, 5, 153, 4, 25, 61, 229, 2, 5, 25, 5, 73, 33, 5, 15, 245, 71, 297, 22, 317, 2, 5, 25, 5, 65, 29, 165, 5, 269, 81, 333, 25, 385, 109, 401, 2, 5, 5, 5, 61, 5, 153, 16, 5, 91, 377, 4, 449, 125, 61, 37, 509, 2

Offset: 1

Scott R. Shannon, Nov 22 2022

The starting point can be any of the 4*n points around the square as changing the starting point simply rotates and/or reflects the resulting pattern formed by the path to one of the four orthogonal directions around the square; this does not change the number of regions formed by the path.
The number of times the path formed by the line touches and leaves the edges of the square is lcm(4*n,k)/k. For k >= n this is the number of points in the star-shaped pattern formed by the path.
The table starts with k = 2 as T(n,1) = 5 for all values of n. The maximum k is 2*n as T(n,2*n + m) = T(n,2*n - m).

			The table begins:
2;
5, 21, 2;
5,  5  4, 61,  2;
5, 29, 5, 73, 25, 105,  2;
5, 25, 5,  5, 31, 141, 11, 157,  2;
5,  5, 5, 85,  5, 153,  4,  25, 61, 229,  2;
5, 25, 5, 73, 33,   5, 15, 245, 71, 297, 22, 317,   2;
5, 25, 5, 65, 29, 165,  5, 269, 81, 333, 25, 385, 109, 401,  2;
5,  5, 5, 61,  5, 153, 16,   5, 91, 377,  4, 449, 125,  61, 37, 509,   2;
5, 25, 5,  5, 25, 137,  5, 285,  5, 385, 31, 501, 141,  25, 11, 613, 169, 629, 2;
.
.
See the attached file for more examples.

Scott R. Shannon, Table for n=1..50.
Scott R. Shannon, Image for T(2,3) = 21.
Scott R. Shannon, Image for T(4,6) = 25.
Scott R. Shannon, Image for T(7,9) = 245.
Scott R. Shannon, Image for T(10,19) = 629.
Scott R. Shannon, Image for T(11,20) = 55.
Scott R. Shannon, Image for T(20,11) = 269.
Scott R. Shannon, Image for T(20,30) = 25.
Scott R. Shannon, Image for T(20,31) = 2277.
Scott R. Shannon, Image for T(50,61) = 11933.

Cf. A358574 (vertices), A358627 (edges), A331452, A355798, A355838, A357058, A358407, A345459.

T(n,k) = A358627(n,k) - A358574(n,k) + 1 by Euler's formula.
T(n,2*n) = 2. The line cuts the square into two parts.
T(n,k) = 5 where n >= 2, k <= n, and k|(4*n). Four lines cut across the square's corners so four additional triangles are created.

A358574 Triangle read by rows: T(n,k) is the number of vertices formed when n points are placed along each edge of a square that divide the edges into n+1 equal parts and a line is continuously drawn from the current point to that k points, 2 <= k <= 2*n, counterclockwise around the square until the starting point is again reached.

Original entry on oeis.org

8, 12, 20, 12, 16, 16, 16, 64, 16, 20, 36, 20, 68, 36, 100, 20, 24, 36, 24, 24, 44, 144, 29, 144, 24, 28, 28, 28, 92, 28, 140, 28, 44, 76, 208, 28, 32, 44, 32, 84, 52, 32, 39, 240, 88, 292, 46, 296, 32, 36, 48, 36, 80, 52, 164, 36, 252, 100, 316, 52, 368, 124, 364, 36, 40, 40, 40, 80, 40, 164, 47, 40, 112, 364, 40, 436, 144, 88, 67, 472, 40

Offset: 1

Scott R. Shannon, Nov 23 2022

See A358556 for further details.

			The table begins:
8;
12, 20, 12;
16, 16, 16, 64, 16;
20, 36, 20, 68, 36, 100, 20;
24, 36, 24, 24, 44, 144, 29, 144, 24;
28, 28, 28, 92, 28, 140, 28,  44,  76, 208, 28;
32, 44, 32, 84, 52,  32, 39, 240,  88, 292, 46, 296,  32;
36, 48, 36, 80, 52, 164, 36, 252, 100, 316, 52, 368, 124, 364, 36;
40, 40, 40, 80, 40, 164, 47,  40, 112, 364, 40, 436, 144,  88, 67, 472, 40;
.
.
See the attached file for more examples.

Scott R. Shannon, Table for n=1..50.
Scott R. Shannon, Image for T(2,3) = 20.
Scott R. Shannon, Image for T(4,6) = 36.
Scott R. Shannon, Image for T(7,9) = 240.
Scott R. Shannon, Image for T(10,19) = 584.
Scott R. Shannon, Image for T(11,20) = 90.
Scott R. Shannon, Image for T(20,11) = 308.
Scott R. Shannon, Image for T(20,30) = 100.
Scott R. Shannon, Image for T(20,31) = 2220.

Cf. A358556 (regions), A358627 (edges), A331452, A355798, A355838, A357058, A358407, A345459.

T(n,k) = A358627(n,k) - A358556(n,k) + 1 by Euler's formula.
T(n,2*n) = 4*(n + 1). The line cuts the square into two parts so no new vertices are created.
T(n,k) = 4*(n + 1) where k <= n and k|(4*n). Four lines cut across the square's corners so no new vertices are created.

A358627 Triangle read by rows: T(n,k) is the number of edges formed when n points are placed along each edge of a square that divide the edges into n+1 equal parts and a line is continuously drawn from the current point to that k points, 2 <= k <= 2*n, counterclockwise around the square until the starting point is again reached.

Original entry on oeis.org

9, 16, 40, 13, 20, 20, 19, 124, 17, 24, 64, 24, 140, 60, 204, 21, 28, 60, 28, 28, 74, 284, 39, 300, 25, 32, 32, 32, 176, 32, 292, 31, 68, 136, 436, 29, 36, 68, 36, 156, 84, 36, 53, 484, 158, 588, 67, 612, 33, 40, 72, 40, 144, 80, 328, 40, 520, 180, 648, 76, 752, 232, 764, 37, 44, 44, 44, 140, 44, 316, 62, 44, 202, 740, 43, 884, 268, 148, 103, 980, 41

Offset: 1

Scott R. Shannon, Nov 24 2022

See A358556 for further details and images of the squares.

			The table begins:
9;
16, 40, 13;
20, 20, 19, 124, 17;
24, 64, 24, 140, 60, 204, 21;
28, 60, 28,  28, 74, 284, 39, 300,  25;
32, 32, 32, 176, 32, 292, 31,  68, 136, 436, 29;
36, 68, 36, 156, 84,  36, 53, 484, 158, 588, 67, 612,  33;
40, 72, 40, 144, 80, 328, 40, 520, 180, 648, 76, 752, 232, 764,  37;
44, 44, 44, 140, 44, 316, 62,  44, 202, 740, 43, 884, 268, 148, 103, 980, 41;
.
.
See the attached file for more examples.

Scott R. Shannon, Table for n=1..50.

Cf. A358556 (regions), A358574 (vertices), A331452, A355798, A355838, A357058, A358407, A345459.

T(n,k) = A358574(n,k) + A358556(n,k) - 1 by Euler's formula.
T(n,2*n) = 4*(n + 1) + 1. The line cuts the square into two parts so one additional edge is created.
T(n,k) = 4*(n + 2) where n >= 2, k <= n, and k|(4*n). Four lines cut across the square's corners so four additional edges are created.

cp's OEIS Frontend

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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A357196 Number of regions 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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A357061 Number of edges 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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A357216 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of regions 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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A357254 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of edges 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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