A355801 -id:A355801 - 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.

A355799 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 the edge on the opposite side of the square.

Original entry on oeis.org

4, 9, 25, 93, 277, 597, 1405, 2421, 4357, 6661, 11261, 14593, 23625, 30121, 41453, 54477, 75985, 87677, 122433, 139461, 177965, 216017, 275733, 298805, 383497, 439909, 522473, 588597, 729501, 763149, 963573, 1045701, 1204481, 1361789, 1546309, 1657125, 2009113, 2166617, 2418733, 2602789

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.

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

a(n) = A355800(n) - A355798(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.

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.

A367665 Irregular table read by rows: T(n,k) is the number of k-sided polygons, for n>=1 and k>=3, in a hexagon 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 hexagon.

Original entry on oeis.org

0, 0, 0, 1, 0, 6, 30, 12, 12, 126, 102, 6, 18, 384, 294, 78, 18, 0, 6, 888, 726, 300, 42, 12, 1824, 1500, 594, 216, 12, 3186, 3048, 1206, 186, 36, 5592, 4914, 1866, 588, 48, 6, 8736, 8298, 3108, 816, 108, 12, 13164, 12786, 5046, 1368, 180, 18, 19578, 17940, 7242, 1854, 144, 60, 26700, 27426, 10842, 2652, 324, 18

Offset: 1

Scott R. Shannon, Nov 26 2023

			The table begins:
0, 0, 0, 1;
0, 6;
30, 12, 12;
126, 102, 6, 18;
384, 294, 78, 18, 0, 6;
888, 726, 300, 42, 12;
1824, 1500, 594, 216, 12;
3186, 3048, 1206, 186, 36;
5592, 4914, 1866, 588, 48, 6;
8736, 8298, 3108, 816, 108, 12;
13164, 12786, 5046, 1368, 180, 18;
19578, 17940, 7242, 1854, 144, 60;
26700, 27426, 10842, 2652, 324, 18;
36972, 37134, 15000, 3462, 408, 72;
50316, 48552, 19398, 4830, 480, 96, 0, 12;
64398, 68388, 27018, 6006, 852, 132;
83052, 89268, 35178, 8526, 1140, 204, 12;
106956, 109938, 42600, 10218, 1116, 204, 24;
131664, 145332, 56622, 14160, 1908, 234, 12;
166338, 173514, 69426, 16716, 2004, 252;
202932, 215028, 82374, 20202, 2400, 324, 12;
.
.

Scott R. Shannon, Image for T(9,k).

Cf. A367662 (regions), A367663 (vertices), A367664 (edges), A355801.

Sum of row(n) = A367662(n).

A359656 Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, 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

0, 1, 0, 4, 56, 40, 368, 300, 48, 12, 3376, 3408, 960, 96, 7536, 7524, 2240, 436, 8, 42048, 45112, 13912, 2868, 168, 28, 97720, 105980, 34496, 7020, 832, 52, 8, 267240, 290456, 100560, 20576, 2688, 160, 24, 461800, 509824, 174400, 36228, 4608, 324, 16, 1411272, 1594296, 569152, 126408, 16856, 1408, 104

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.

See A359653 and A359654 for images of the square.

			The table begins:
0, 1;
0, 4;
56, 40;
368, 300, 48, 12;
3376, 3408, 960, 96;
7536, 7524, 2240, 436, 8;
42048, 45112, 13912, 2868, 168, 28;
97720, 105980, 34496, 7020, 832, 52, 8;
267240, 290456, 100560, 20576, 2688, 160, 24;
461800, 509824, 174400, 36228, 4608, 324, 16;
1411272, 1594296, 569152, 126408, 16856, 1408, 104;
2054616, 2300184, 830280, 184664, 24480, 2332, 128, 8;
5296752, 6001228, 2253456, 517564, 72888, 7532, 472, 4;
.
.

Cf. A359653 (regions), A359654 (vertices), A359655 (edges), A005728, A358889, A358885, A355801, A358951, A006842, A006843.

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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A355799 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 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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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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A367665 Irregular table read by rows: T(n,k) is the number of k-sided polygons, for n>=1 and k>=3, in a hexagon 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 hexagon.

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