A355949 -id:A355949 - cp's OEIS frontend

A108914 Number of regions formed inside square by diagonals and the segments joining the vertices to the points dividing the sides into n equal length segments.

4, 32, 96, 188, 332, 460, 712, 916, 1204, 1488, 1904, 2108, 2716, 3080, 3532, 4068, 4772, 5140, 6016, 6392, 7188, 7992, 8936, 9260, 10484, 11312, 12208, 12968, 14396, 14660, 16504, 17220, 18436, 19680, 20756, 21548, 23692, 24728, 25992, 26868, 29204, 29704, 32176, 33068, 34444, 36552, 38552

Offset: 1

Len Smiley and Brian Wick ( mathclub(AT)math.uaa.alaska.edu ), Jul 19 2005

nonn

Scott R. Shannon, Table of n, a(n) for n = 1..100
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.
Scott R. Shannon, Image for n = 30.
L. Smiley, The case n=6. Note 3- and 4-fold off-diagonal concurrencies.
L. Smiley, The case n=7. Note there are no off-diagonal concurrencies.

A092098 is the corresponding count for triangles.

A355949 (vertices), A355948 (edges), A355992 (k-gons), A355838, A355798.

If n=1 or n is prime, a(n)=18*n^2-26*n+12.
If n is composite, vanishing regions from 3- and 4-fold concurrency must be subtracted.
a(n) = A355948(n) - A355949(n) + 1 by Euler's formula.

a(23), a(33) corrected, a(41) and above by Scott R. Shannon, Jul 22 2022

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.

A355948 Number of edges 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

8, 56, 176, 344, 632, 840, 1376, 1736, 2312, 2840, 3728, 3968, 5336, 5960, 6816, 7880, 9416, 9912, 11888, 12344, 14008, 15656, 17696, 17872, 20648, 22232, 23984, 25304, 28568, 28376, 32768, 33800, 36296, 38840, 40848, 42008, 47096, 48872, 51296, 52568, 58088, 58136, 64016, 65144, 67712, 72392

Offset: 1

Scott R. Shannon, Jul 21 2022

nonn

See A108914 for images of the squares.

Cf. A108914 (regions), A355949 (vertices), A355992 (k-gons), A355840, A331452, A335678.

a(n) = A108914(n) + A355949(n) - 1 by Euler's formula.

A355992 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 four corner vertices to the points dividing the sides into n equal parts.

Original entry on oeis.org

4, 24, 8, 56, 28, 12, 96, 80, 8, 4, 144, 140, 36, 12, 216, 216, 24, 4, 272, 332, 76, 24, 8, 360, 448, 80, 28, 456, 572, 132, 36, 8, 568, 728, 128, 64, 656, 916, 260, 28, 40, 4, 792, 1104, 176, 36, 928, 1308, 316, 128, 32, 4, 1064, 1568, 304, 128, 16, 1240, 1772, 396, 88, 32, 4, 1416, 2032, 432, 156, 32

Offset: 1

Scott R. Shannon, Jul 22 2022

Up to n = 100 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 A108914 for more images of the square.

			The table begins:
4;
24,   8;
56,   28,   12;
96,   80,   8,   4;
144,  140,  36,  12;
216,  216,  24,  4;
272,  332,  76,  24,  8;
360,  448,  80,  28;
456,  572,  132, 36,  8;
568,  728,  128, 64;
656,  916,  260, 28,  40, 4;
792,  1104, 176, 36;
928,  1308, 316, 128, 32, 4;
1064, 1568, 304, 128, 16;
1240, 1772, 396, 88,  32, 4;
1416, 2032, 432, 156, 32;
.
.

Scott R. Shannon, Image for n = 13.

Cf. A108914 (regions), A355948 (edges), A355949 (vertices), A355841, A331452, A335678.

A354544 Table read by antidiagonals: T(n,k) (n >= 3, k >= 1) is the number of vertices formed in a regular n-gon by straight line segments when connecting the n corner vertices to the points dividing the sides into k equal parts.

Original entry on oeis.org

3, 7, 5, 21, 25, 10, 25, 81, 61, 19, 63, 157, 285, 205, 42, 67, 301, 476, 541, 358, 57, 129, 381, 1020, 1327, 1526, 681, 135, 133, 665, 1311, 2185, 2682, 2417, 1234, 171, 219, 821, 2215, 3067, 5250, 5073, 4716, 2131, 341, 223, 1109, 2666, 4921, 7246, 8937, 8623, 6861, 3169, 313

Offset: 3

Scott R. Shannon, Aug 18 2022

			The table begins:
3,    7,     21,     25,     63,     67,     129,    133,     219,     223,...
5,    25,    81,     157,    301,    381,    665,    821,     1109,    1353,...
10,   61,    285,    476,    1020,   1311,   2215,   2666,    3810,    4421,...
19,   205,   541,    1327,   2185,   3067,   4921,   6739,    8401,    10507,...
42,   358,   1526,   2682,   5250,   7246,   11214,  14050,   19418,   23094,...
57,   681,   2417,   5073,   8937,   13089,  19473,  26049,   33769,   42497,...
135,  1234,  4716,   8623,   16173,  22780,  34398,  43813,   59391,   71614,...
171,  2131,  6861,   14271,  24731,  36161,  53071,  70751,   91761,   115001,...
341,  3169,  11451,  21143,  38665,  55221,  81983,  105403,  141405,  171689,...
313,  4837,  14653,  31009,  53989,  78589,  115909, 154105,  199777,  249961,...
728,  6787,  23491,  43850,  78858,  113517, 166829, 215788,  287404,  350663,...
771,  9927,  30479,  61951,  105155, 157851, 224043, 299727,  386807,  485367,...
1380, 12856, 43080,  81136,  144180, 208636, 304500, 395536,  524040,  641656,...
1393, 17633, 54001,  109265, 184785, 277745, 392737, 525169,  677729,  849249,...
2397, 22288, 73066,  138177, 243355, 353686, 513264, 668815,  882793,  1083564,...
1855, 27595, 88291,  177085, 302167, 450469, 641539, 855829,  1106119, 1384183,...
3895, 36139, 116337, 220933, 386403, 563351, 814093, 1063393, 1399407, 1721059,...
.
.
See the attached text file for more examples and the cross references for further images.

Scott R. Shannon, Image of T(4,9) = 1109.
Scott R. Shannon, Image of T(5,7) = 2215.
Scott R. Shannon, Image of T(7,6) = 7246.
Scott R. Shannon, Image of T(10,3) = 6861.
Scott R. Shannon, Image of T(11,4) = 21143.
Scott R. Shannon, Table for n=3..25.

Cf. A356044 (number of regions), A007569 (first column), A331782 (first row), A355949 (second row).

T(n,1) = A007569(n).
T(3,k) = A331782(k).
T(4,k) = A355949(k).

cp's OEIS Frontend

A108914 Number of regions formed inside square by diagonals and the segments joining the vertices to the points dividing the sides into n equal length segments.

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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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A355948 Number of edges 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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A355992 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 four corner vertices to the points dividing the sides into n equal parts.

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A354544 Table read by antidiagonals: T(n,k) (n >= 3, k >= 1) is the number of vertices formed in a regular n-gon by straight line segments when connecting the n corner vertices to the points dividing the sides into k equal parts.

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