A355839 -id:A355839 - cp's OEIS frontend

A355838 Number of regions 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.

4, 40, 184, 496, 1240, 2144, 4380, 6720, 10860, 15528, 24300, 30152, 46036, 57496, 75056, 96416, 129052, 148512, 198392, 225240, 279576, 336272, 415988, 453376, 565052, 648008, 754808, 848664, 1026040, 1085536, 1331532, 1452704, 1652684, 1862600, 2084888, 2247568, 2662092, 2887944, 3193744

Offset: 1

Scott R. Shannon, Jul 18 2022

nonn

This sequence is similar to A355798 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 = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 16.

Cf. A355839 (vertices), A355840 (edges), A355841 (k-gons), A355798 (without corner vertices), A290131, A331452, A335678.

a(n) = A355840(n) - A355839(n) + 1 by Euler's formula.

A355949 Number of vertices formed in a square by straight line segments when connecting the four corner vertices to the points dividing the sides into n equal parts.

Original entry on oeis.org

5, 25, 81, 157, 301, 381, 665, 821, 1109, 1353, 1825, 1861, 2621, 2881, 3285, 3813, 4645, 4773, 5873, 5953, 6821, 7665, 8761, 8613, 10165, 10921, 11777, 12337, 14173, 13717, 16265, 16581, 17861, 19161, 20093, 20461, 23405, 24145, 25305, 25701, 28885, 28433, 31841, 32077, 33269, 35841, 38185

Offset: 1

Scott R. Shannon, Jul 21 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 = 11.

Cf. A108914 (regions), A355948 (edges), A355992 (k-gons), A355839, A331452, A335678.

a(n) = A355948(n) - A108914(n) + 1 by Euler's formula.

A355840 Number of edges 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

8, 64, 316, 852, 2252, 3780, 8140, 12280, 20172, 28592, 45988, 55508, 87588, 107652, 141060, 181312, 246844, 278352, 380108, 424096, 530764, 638564, 799148, 854448, 1082244, 1235048, 1442572, 1612088, 1975908, 2051784, 2565956, 2773616, 3164916, 3566256, 3997652, 4271136, 5137452, 5537756

Offset: 1

Scott R. Shannon, Jul 18 2022

nonn

This sequence is similar to A355800 but here the corner vertices of the square are also connected to points on the opposite edge.

See A355838 for images of the squares.

Cf. A355838 (regions), A355839 (vertices), A355841 (k-gons), A355800 (without corner vertices), A290131, A331452, A335678.

a(n) = A355838(n) + A355839(n) - 1 by Euler's formula.

A355841 Irregular table read by rows: T(n,k) is the number of k-sided polygons formed, for k>=3, in a square when straight line segments connect 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

4, 40, 128, 44, 12, 320, 152, 24, 616, 512, 84, 28, 1240, 744, 120, 40, 1936, 1928, 372, 136, 8, 3288, 2656, 616, 160, 4960, 4500, 1020, 332, 48, 7224, 6472, 1424, 392, 16, 9760, 11064, 2564, 824, 72, 16, 14144, 12424, 2696, 856, 32, 18312, 20604, 5308, 1468, 328, 16, 24384, 25392, 5968, 1584, 160, 8

Offset: 1

Scott R. Shannon, Jul 18 2022

This sequence is similar to A355801 but here the corner vertices of the square are also connected to points on the opposite edge.
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 A355838 for more images of the square.

			The table begins:
4;
40;
128,   44,    12;
320,   152,   24;
616,   512,   84,    28;
1240,  744,   120,   40;
1936,  1928,  372,   136,  8;
3288,  2656,  616,   160;
4960,  4500,  1020,  332,  48;
7224,  6472,  1424,  392,  16;
9760,  11064, 2564,  824,  72,  16;
14144, 12424, 2696,  856,  32;
18312, 20604, 5308,  1468, 328, 16;
24384, 25392, 5968,  1584, 160, 8;
31816, 32768, 7564,  2652, 240, 16;
40456, 42240, 10384, 3064, 248, 24;
49384, 59152, 15068, 4680, 704, 64;
.
.

Scott R. Shannon, Image for n = 10.

Cf. A355838 (regions), A355839 (vertices), A355840 (edges), A355801 (without corner vertices), A290131, A331452, A335678.

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.

cp's OEIS Frontend

A355838 Number of regions 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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A355949 Number of vertices formed in a square by straight line segments when connecting the four corner vertices to the points dividing the sides into n equal parts.

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A355840 Number of edges 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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A355841 Irregular table read by rows: T(n,k) is the number of k-sided polygons formed, for k>=3, in a square when straight line segments connect 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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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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