A355799 -id:A355799 - cp's OEIS frontend

A355798 Number of regions formed in a square by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.

1, 4, 24, 104, 316, 712, 1588, 2816, 4940, 7672, 12444, 16840, 25968, 34088, 46260, 61048, 82792, 98984, 133032, 156072, 196236, 239048, 298292, 334032, 417072, 483856, 570200, 649816, 786412, 850000, 1037628, 1145424, 1311536, 1485880, 1677660, 1828360, 2158192, 2357376, 2623604, 2852688

Offset: 1

Scott R. Shannon, Jul 17 2022

nonn

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.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 16.
Scott R. Shannon, Alternative version for n = 15. The lines joining the vertices that are exactly opposite are omitted, forming a more carpet-like image. The number of regions formed, 42224, is less than the sequence version, although more 5,6,7 and 8-gon regions are present.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, pp. 20-21.

Cf. A355799 (vertices), A355800 (edges), A355801 (k-gons), A255011 (all vertices), A290131, A331452, A335678.

a(n) = A355800(n) - A355799(n) + 1 by Euler's formula.

A355800 Number of edges formed in a square by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.

Original entry on oeis.org

4, 12, 48, 196, 592, 1308, 2992, 5236, 9296, 14332, 23704, 31432, 49592, 64208, 87712, 115524, 158776, 186660, 255464, 295532, 374200, 455064, 574024, 632836, 800568, 923764, 1092672, 1238412, 1515912, 1613148, 2001200, 2191124, 2516016, 2847668, 3223968, 3485484, 4167304, 4523992, 5042336

Offset: 1

Scott R. Shannon, Jul 17 2022

nonn

See A355798 for images of the squares.

Cf. A355798 (regions), A355799 (vertices), A355801 (k-gons), A255011 (all vertices), A290131, A331452, A335678.

a(n) = A355798(n) + A355799(n) - 1 by Euler's formula.

A355801 Irregular table read by rows: T(n,k) is the number of k-sided polygons, for k>=3, in a square when straight line segments connect the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.

Original entry on oeis.org

0, 1, 0, 4, 12, 12, 56, 32, 16, 156, 124, 24, 8, 0, 4, 384, 228, 72, 28, 716, 648, 144, 68, 8, 4, 1312, 1144, 240, 112, 8, 2244, 1912, 528, 256, 3528, 3072, 696, 360, 16, 5012, 5536, 1296, 524, 48, 28, 7696, 6596, 1960, 572, 16, 10340, 11448, 2968, 1028, 160, 24, 14520, 14428, 3872, 1156, 104, 8

Offset: 1

Scott R. Shannon, Jul 17 2022

Up to n = 50 the maximum sided k-gon created is the 8-gon. It is plausible this is the maximum sided k-gon for all n, although this is unknown.
See A355798 for more images of the square.
The keyword "look" is for the n = 10 image. - N. J. A. Sloane, Jul 21 2022

			The table begins:
0,     1;
0,     4;
12,    12;
56,    32,    16;
156,   124,   24,   8,    0,   4;
384,   228,   72,   28;
716,   648,   144,  68,   8,   4;
1312,  1144,  240,  112,  8;
2244,  1912,  528,  256;
3528,  3072,  696,  360,  16;
5012,  5536,  1296, 524,  48,  28;
7696,  6596,  1960, 572,  16;
10340, 11448, 2968, 1028, 160, 24;
14520, 14428, 3872, 1156, 104, 8;
19588, 19156, 5296, 2052, 160, 8;
25392, 26112, 7160, 2152, 208, 24;
31820, 37244, 9936, 3240, 488, 64;
.
.

Scott R. Shannon, Image for n = 10.

Cf. A355798 (regions), A355799 (vertices), A355800 (edges), A355801 (k-gons), A255011 (all vertices), A290131, A331452, A335678.

A355839 Number of vertices formed in a square by straight line segments when connecting the n+1 points along each edge that divide it into n equal parts to the n+1 points on the edge on the opposite side of the square.

Original entry on oeis.org

5, 25, 133, 357, 1013, 1637, 3761, 5561, 9313, 13065, 21689, 25357, 41553, 50157, 66005, 84897, 117793, 129841, 181717, 198857, 251189, 302293, 383161, 401073, 517193, 587041, 687765, 763425, 949869, 966249, 1234425, 1320913, 1512233, 1703657, 1912765, 2023569, 2475361, 2649813, 2934997

Offset: 1

Scott R. Shannon, Jul 18 2022

nonn

This sequence is similar to A355799 but here the corner vertices of the square are also connected to points on the opposite edge.

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.
Scott R. Shannon, Image for n = 11.

Cf. A355838 (regions), A355840 (edges), A355841 (k-gons), A355799 (without corner vertices), A290131, A331452, A335678.

a(n) = A355840(n) - A355838(n) + 1 by Euler's formula.

A359654 Number of vertices formed in a square with edge length 1 by straight line segments when connecting the internal edge points that divide the sides into segments with lengths equal to the Farey series of order n to the equivalent points on the opposite side of the square.

Original entry on oeis.org

4, 9, 77, 593, 6749, 15569, 93281, 222933, 623409, 1087393, 3453289, 5011009, 13271517

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 10 2023

The number of points internal to each edge is given by A005728(n) - 2.

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. A359653 (regions), A359655 (edges), A359656 (k-gons), A005728, A358887, A358883, A355799, A358949, A006842, A006843.

a(n) = A359655(n) - A359653(n) + 1 by Euler's formula.

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.

A367663 Number of vertices formed in a hexagon by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the hexagon.

Original entry on oeis.org

6, 13, 55, 223, 673, 1753, 3787, 6937, 11851, 19297, 30211, 42877, 63247, 86323, 114049, 155701, 204169, 251293, 329989, 400081, 487537, 607189, 742747, 858241, 1041835, 1231789, 1424647, 1667659, 1968427, 2165947, 2600779, 2939863, 3286159, 3784153, 4217395, 4687771, 5688804, 6316332

Offset: 1

Scott R. Shannon, Nov 26 2023

Keyword "look" is because of the linked images. - N. J. A. Sloane, Dec 01 2023

Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 9.

Cf. A367662 (regions), A367664 (edges), A367665 (k-gons), A355799.

a(n) = A367664(n) - A367662(n) + 1 by Euler's formula.

A358408 Number of vertices formed in a square by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on each of the two adjacent edges of the square.

Original entry on oeis.org

4, 8, 32, 144, 468, 1160, 2512, 4836, 8468, 13760, 21784, 31168, 46596, 64760, 86912, 114656, 154400, 194116, 253160, 310408, 382712, 469296, 580688, 677656, 822928, 969880, 1141112, 1319984, 1566512, 1755032, 2080376, 2349188, 2686452, 3052184, 3450348, 3800756, 4387404, 4880560, 5443192

Offset: 1

Scott R. Shannon, Nov 14 2022

nonn

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 = 10.

Cf. A358407 (regions), A358409 (edges), A355799, A331449.

a(n) = A358409(n) - A358407(n) + 1 by Euler's formula.

cp's OEIS Frontend

A355798 Number of regions formed in a square by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.

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A355800 Number of edges formed in a square by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.

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A355801 Irregular table read by rows: T(n,k) is the number of k-sided polygons, for k>=3, in a square when straight line segments connect the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.

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A355839 Number of vertices formed in a square by straight line segments when connecting the n+1 points along each edge that divide it into n equal parts to the n+1 points on the edge on the opposite side of the square.

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A359654 Number of vertices formed in a square with edge length 1 by straight line segments when connecting the internal edge points that divide the sides into segments with lengths equal to the Farey series of order n to the equivalent points on the opposite side of the square.

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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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A367663 Number of vertices formed in a hexagon by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the hexagon.

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A358408 Number of vertices formed in a square by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on each of the two adjacent edges of the square.

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