A335678 -id:A335678 - cp's OEIS frontend

A335682 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of simple interior vertices in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 6, 6, 0, 0, 10, 12, 12, 10, 0, 0, 15, 18, 24, 18, 15, 0, 0, 21, 27, 36, 36, 27, 21, 0, 0, 28, 36, 54, 54, 54, 36, 28, 0, 0, 36, 48, 72, 82, 82, 72, 48, 36, 0, 0, 45, 60, 96, 108, 124, 108, 96, 60, 45, 0, 0, 55, 75, 120, 144, 163, 163, 144, 120, 75, 55, 0

Offset: 1

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Jun 28 2020

A simple interior vertex is a vertex where exactly two lines cross.  In graph theory terms, this is an interior vertex of degree 4.
The case m=n (the main diagonal) is dealt with in A334701. A306302 has illustrations for the diagonal case for m = 1 to 15.
Also A335678 has colored illustrations for many values of m and n.
This is the only one of the five arrays (A335678-A335682) that does not have an explicit formula.
Let G_m(x) = g.f. for row m. For m <= 9, G_m appears to be a rational function of x with denominator D_m(x), where (writing C_k for the k-th cyclotomic polynomial):
D_3 = D_4 = C_1^3*C_2
D_5 = C_1^3*C_2*C_4
D_6 = C_1^3*C_2*C_4*C_5
D_7 = C_1^3*C_2*C_3*C_4*C_5*C_6
D_8 = D_9 = C_1^3*C_2*C_3*C_4*C_5*C_6*C_7

			The initial rows of the array are:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...
0, 3, 6, 12, 18, 27, 36, 48, 60, 75, 90, 108, ...
0, 6, 12, 24, 36, 54, 72, 96, 120, 150, 180, 216, ...
0, 10, 18, 36, 54, 82, 108, 144, 180, 226, 270, 324, ...
0, 15, 27, 54, 82, 124, 163, 217, 272, 342, 408, 489, ...
0, 21, 36, 72, 108, 163, 214, 286, 358, 451, 536, 642, ...
0, 28, 48, 96, 144, 217, 286, 382, 478, 602, 715, 856, ...
0, 36, 60, 120, 180, 272, 358, 478, 598, 754, 894, 1070, ...
0, 45, 75, 150, 226, 342, 451, 602, 754, 950, 1126, 1347, ...
0, 55, 90, 180, 270, 408, 536, 715, 894, 1126, 1334, 1597, ...
0, 66, 108, 216, 324, 489, 642, 856, 1070, 1347, 1597, 1912, ...
...
The initial antidiagonals are:
0
0, 0
0, 1, 0
0, 3, 3, 0
0, 6, 6, 6, 0
0, 10, 12, 12, 10, 0
0, 15, 18, 24, 18, 15, 0
0, 21, 27, 36, 36, 27, 21, 0
0, 28, 36, 54, 54, 54, 36, 28, 0
0, 36, 48, 72, 82, 82, 72, 48, 36, 0
0, 45, 60, 96, 108, 124, 108, 96, 60, 45, 0
0, 55, 75, 120, 144, 163, 163, 144, 120, 75, 55, 0
...

Lars Blomberg, Table of n, a(n) for n = 1..9870 (the first 140 antidiagonals)
Index entries for sequences related to stained glass windows

This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.

For the diagonal case see A306302, A331755, A334701.

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.

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.

cp's OEIS Frontend

A335682 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of simple interior vertices in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

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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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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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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.

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