A335679 -id:A335679 - cp's OEIS frontend

A335678 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of cells 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, 1, 1, 2, 4, 2, 3, 8, 8, 3, 4, 13, 16, 13, 4, 5, 19, 27, 27, 19, 5, 6, 26, 40, 46, 40, 26, 6, 7, 34, 56, 69, 69, 56, 34, 7, 8, 43, 74, 98, 104, 98, 74, 43, 8, 9, 53, 95, 130, 149, 149, 130, 95, 53, 9, 10, 64, 118, 168, 198, 214, 198, 168, 118, 64, 10, 11, 76, 144, 210, 257, 285, 285, 257, 210, 144, 76, 11

Offset: 1

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

The case m=n (the main diagonal) is dealt with in A306302, where there are illustrations for m = 1 to 15.

			The initial rows of the array are:
  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...
  1, 4, 8, 13, 19, 26, 34, 43, 53, 64, 76, 89, ...
  2, 8, 16, 27, 40, 56, 74, 95, 118, 144, 172, 203, ...
  3, 13, 27, 46, 69, 98, 130, 168, 210, 257, 308, 365, ...
  4, 19, 40, 69, 104, 149, 198, 257, 322, 395, 474, 563, ...
  5, 26, 56, 98, 149, 214, 285, 371, 466, 573, 688, 818, ...
  6, 34, 74, 130, 198, 285, 380, 496, 624, 768, 922, 1097, ...
  7, 43, 95, 168, 257, 371, 496, 648, 816, 1005, 1207, 1437, ...
  8, 53, 118, 210, 322, 466, 624, 816, 1028, 1267, 1522, 1813, ...
  9, 64, 144, 257, 395, 573, 768, 1005, 1267, 1562, 1877, 2237, ...
  10, 76, 172, 308, 474, 688, 922, 1207, 1522, 1877, 2256, 2690, ...
  ...
The initial antidiagonals are:
  0
  1, 1
  2, 4, 2
  3, 8, 8, 3
  4, 13, 16, 13, 4
  5, 19, 27, 27, 19, 5
  6, 26, 40, 46, 40, 26, 6
  7, 34, 56, 69, 69, 56, 34, 7
  8, 43, 74, 98, 104, 98, 74, 43, 8
  9, 53, 95, 130, 149, 149, 130, 95, 53, 9
  10, 64, 118, 168, 198, 214, 198, 168, 118, 64, 10
  ...

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090.
M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5.
S. Legendre, The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph, J. Integer Seqs., Vol. 12, 2009.
Scott R. Shannon, Colored illustration for T(2,1)
Scott R. Shannon, Colored illustration for T(2,2)
Scott R. Shannon, Colored illustration for T(3,1)
Scott R. Shannon, Colored illustration for T(3,2)
Scott R. Shannon, Colored illustration for T(3,3)
Scott R. Shannon, Colored illustration for T(4,1)
Scott R. Shannon, Colored illustration for T(4,2)
Scott R. Shannon, Colored illustration for T(4,3)
Scott R. Shannon, Colored illustration for T(4,4)
Scott R. Shannon, Colored illustration for T(5,1)
Scott R. Shannon, Colored illustration for T(5,2)
Scott R. Shannon, Colored illustration for T(5,3)
Scott R. Shannon, Colored illustration for T(5,4)
Scott R. Shannon, Colored illustration for T(5,5)
Scott R. Shannon, Colored illustration for T(6,2)
Scott R. Shannon, Colored illustration for T(6,3)
Scott R. Shannon, Colored illustration for T(6,4)
Scott R. Shannon, Colored illustration for T(6,5)
Scott R. Shannon, Colored illustration for T(6,6)
Scott R. Shannon, Colored illustration for T(7,1)
Scott R. Shannon, Colored illustration for T(7,2)
Scott R. Shannon, Colored illustration for T(7,3)
Scott R. Shannon, Colored illustration for T(7,4)
Scott R. Shannon, Colored illustration for T(7,5)
Scott R. Shannon, Colored illustration for T(7,6)
Scott R. Shannon, Colored illustration for T(7,7)
Scott R. Shannon, Colored illustration for T(8,1)
Scott R. Shannon, Colored illustration for T(8,2)
Scott R. Shannon, Colored illustration for T(8,3)
Scott R. Shannon, Colored illustration for T(8,4)
Scott R. Shannon, Colored illustration for T(8,5)
Scott R. Shannon, Colored illustration for T(8,6)
Scott R. Shannon, Colored illustration for T(8,7)
Scott R. Shannon, Colored illustration for T(8,8)
Scott R. Shannon, Colored illustration for T(14,7)
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 see A306302.
See also A114999.

Euler's formula implies that A335679[m,n] = A335678[m,n] + A335680[m,n] - 1 for all m,n.
Comment from Max Alekseyev, Jun 28 2020 (Start):
T(m,n) = A114999(m-1,n-1) + m*n - 1 for all m, n >= 1. This follows from the Alekseyev-Basova-Zolotykh (2015) article.
Proof: Here is the appropriate modification of the corresponding comment in A306302: Assuming that K(m,n) has vertices at (i,0) and (j,1), for i=0..m-1 and j=0..n-1, the projective map (x,y) -> ((1-y)/(x+1), y/(x+1)) maps K(m,n) to the partition of the right triangle described by Alekseyev et al. (2015), for which Theorem 13 gives the number of regions, line segments, and intersection points. (End)
Max Alekseyev's formula is an analog of Theorem 3 of Griffiths (2010), and gives an explicit formula for this array. - N. J. A. Sloane, Jun 30 2020

A335680 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of 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.

Original entry on oeis.org

2, 3, 3, 4, 5, 4, 5, 8, 8, 5, 6, 12, 13, 12, 6, 7, 17, 21, 21, 17, 7, 8, 23, 30, 35, 30, 23, 8, 9, 30, 42, 51, 51, 42, 30, 9, 10, 38, 55, 73, 75, 73, 55, 38, 10, 11, 47, 71, 96, 109, 109, 96, 71, 47, 11, 12, 57, 88, 125, 143, 159, 143, 125, 88, 57, 12, 13, 68, 108, 156, 187, 209, 209, 187, 156, 108, 68, 13

Offset: 1

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

The case m=n (the main diagonal) is dealt with in A331755. A306302 has illustrations for the diagonal case for m = 1 to 15.

Also A335678 has colored illustrations for many values of m and n.

			The initial rows of the array are:
  2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ...
  3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, ...
  4, 8, 13, 21, 30, 42, 55, 71, 88, 108, 129, 153, ...
  5, 12, 21, 35, 51, 73, 96, 125, 156, 192, 230, 274, ...
  6, 17, 30, 51, 75, 109, 143, 187, 234, 289, 346, 413, ...
  7, 23, 42, 73, 109, 159, 209, 274, 344, 426, 510, 609, ...
  8, 30, 55, 96, 143, 209, 275, 362, 455, 564, 674, 805, ...
  9, 38, 71, 125, 187, 274, 362, 477, 600, 744, 889, 1062, ...
  10, 47, 88, 156, 234, 344, 455, 600, 755, 937, 1119, 1337, ...
  11, 57, 108, 192, 289, 426, 564, 744, 937, 1163, 1389, 1660, ...
  12, 68, 129, 230, 346, 510, 674, 889, 1119, 1389, 1659, 1984, ...
  ...
The initial antidiagonals are:
  2
  3, 3
  4, 5, 4
  5, 8, 8, 5
  6, 12, 13, 12, 6
  7, 17, 21, 21, 17, 7
  8, 23, 30, 35, 30, 23, 8
  9, 30, 42, 51, 51, 42, 30, 9
  10, 38, 55, 73, 75, 73, 55, 38, 10
  11, 47, 71, 96, 109, 109, 96, 71, 47, 11
  12, 57, 88, 125, 143, 159, 143, 125, 88, 57, 12
  ...

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090.
M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5.
S. Legendre, The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph, J. Integer Seqs., Vol. 12, 2009.
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 and A331755.
Cf. A114999, A331762.

Comment from Max Alekseyev, Jun 28 2020 (Start):
T(m,n) = A114999(m-1,n-1) - A331762(m-1,n-1) + m + n for all m, n >= 1. This follows from the Alekseyev-Basova-Zolotykh (2015) article.
Proof: Here is the appropriate modification of the corresponding comment in A306302: Assuming that K(m,n) has vertices at (i,0) and (j,1), for i=0..m-1 and j=0..n-1, the projective map (x,y) -> ((1-y)/(x+1), y/(x+1)) maps K(m,n) to the partition of the right triangle described by Alekseyev et al. (2015), for which Theorem 13 gives the number of regions, line segments, and intersection points. (End)
Max Alekseyev's formula is an analog of Proposition 9 of Legendre (2009), and gives an explicit formula for this array. - N. J. A. Sloane, Jun 30 2020

A335681 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of 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.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 7, 6, 0, 0, 10, 14, 14, 10, 0, 0, 15, 22, 27, 22, 15, 0, 0, 21, 33, 42, 42, 33, 21, 0, 0, 28, 45, 63, 65, 63, 45, 28, 0, 0, 36, 60, 85, 98, 98, 85, 60, 36, 0, 0, 45, 76, 113, 131, 147, 131, 113, 76, 45, 0, 0, 55, 95, 143, 174, 196, 196, 174, 143, 95, 55, 0

Offset: 1

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

The case m=n (the main diagonal) is dealt with in A331755. A306302 has illustrations for the diagonal case for m = 1 to 15.

Also A335678 has colored illustrations for many values of m and n.

			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, 7, 14, 22, 33, 45, 60, 76, 95, 115, 138, ...
  0, 6, 14, 27, 42, 63, 85, 113, 143, 178, 215, 258, ...
  0, 10, 22, 42, 65, 98, 131, 174, 220, 274, 330, 396, ...
  0, 15, 33, 63, 98, 147, 196, 260, 329, 410, 493, 591, ...
  0, 21, 45, 85, 131, 196, 261, 347, 439, 547, 656, 786, ...
  0, 28, 60, 113, 174, 260, 347, 461, 583, 726, 870, 1042, ...
  0, 36, 76, 143, 220, 329, 439, 583, 737, 918, 1099, 1316, ...
  0, 45, 95, 178, 274, 410, 547, 726, 918, 1143, 1368, 1638, ...
  0, 55, 115, 215, 330, 493, 656, 870, 1099, 1368, 1637, 1961, ...
  ...
The initial antidiagonals are:
  0
  0, 0
  0, 1, 0
  0, 3, 3, 0
  0, 6, 7, 6, 0
  0, 10, 14, 14, 10, 0
  0, 15, 22, 27, 22, 15, 0
  0, 21, 33, 42, 42, 33, 21, 0
  0, 28, 45, 63, 65, 63, 45, 28, 0
  0, 36, 60, 85, 98, 98, 85, 60, 36, 0
  0, 45, 76, 113, 131, 147, 131, 113, 76, 45, 0
  0, 55, 95, 143, 174, 196, 196, 174, 143, 95, 55, 0
  ...

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 and A331755.

It follows from the definitions that T(m,n) = A335680(m,n) - m - n. Note that there is an explicit formula for the latter sequence.

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.

Original entry on oeis.org

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.

cp's OEIS Frontend

A335678 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of cells 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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A335680 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of 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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Comments

Examples

Links

Crossrefs

Formula

A335681 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of 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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Comments

Examples

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs