A331755 -id:A331755 - cp's OEIS frontend

A333280 Column 2 of triangle in A333278.

28, 92, 296, 652, 1408, 2470, 4312, 6774, 10428, 14992, 21492, 29328, 39876, 52184, 67616, 85588, 108192, 133674, 164992, 200158, 241560, 287428, 341768, 401472, 470764, 546230, 632404, 726170, 833420, 948550, 1079204, 1220054, 1376552, 1543742, 1729000

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 15 2020

nonn

For the graphs defined in A331452 and A288187 only the counts for graphs that are one square wide have formulas for regions, edges, and vertices (see A306302, A331757, A331755). For width 2 there are six such sequences (A331766, A331765, A331763; A333279, A333280, A333281). It would be nice to have a formula for any one of them.

See A333279 for illustrations.

Lars Blomberg, Table of n, a(n) for n = 1..200
Hugo Pfoertner, Illustrations of Chamber Complexes up to 5 X 5.

Cf. A331452, A288187; A331766, A331765, A331763; A333279, A333280, A333281.

a(10) and beyond from Lars Blomberg, May 23 2021

A333281 Column 2 of triangle in A288180.

Original entry on oeis.org

13, 37, 121, 265, 587, 1019, 1797, 2823, 4369, 6257, 9001, 12289, 16775, 21905, 28383, 35901, 45463, 56119, 69351, 84167, 101687, 120869, 143777, 168873, 198191, 229771, 266015, 305379, 350673, 399035, 454243, 513619, 579787, 649899, 727927, 810907, 903581

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 15 2020

nonn

For the graphs defined in A331452 and A288187 only the counts for graphs that are one square wide have formulas for regions, edges, and vertices (see A306302, A331757, A331755). For width 2 there are six such sequences (A331766, A331765, A331763; A333279, A333280, A333281). It would be nice to have a formula for any one of them.

See A333279 for illustrations.

Lars Blomberg, Table of n, a(n) for n = 1..200
Hugo Pfoertner, Illustrations of Chamber Complexes up to 5 X 5.

Cf. A331452, A288187; A331766, A331765, A331763; A333279, A333280, A333281.

a(10) and beyond from Lars Blomberg, May 23 2021

A333276 a(n) = Sum_k k*A333274(n,k).

Original entry on oeis.org

12, 50, 152, 346, 732, 1294, 2232, 3546, 5428, 7806, 11136, 15226, 20676, 27150, 35048, 44386, 56044, 69302, 85480, 103882, 125180, 148942, 176968, 208034, 243772, 283014, 327272, 375826, 431212, 490918, 558456, 631978, 712844, 799726, 895152, 997322, 1110628

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 15 2020

nonn

a(n)/A331755(n) is the average number of polygons touching a vertex in the graph defined in A306302.

Lars Blomberg, Table of n, a(n) for n = 1..100
Scott R. Shannon, Images of vertices for n=1.
Scott R. Shannon, Images of vertices for n=2.
Scott R. Shannon, Images of vertices for n=3.
Scott R. Shannon, Images of vertices for n=4.
Scott R. Shannon, Images of vertices for n=9.
Scott R. Shannon, Images of vertices for n=14.

Cf. A159065, A306302, A333274, A333275, A333277.

a(15) and beyond from Lars Blomberg, Jun 17 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.

A347750 Number of intersection points when every pair of vertices of a row of n adjacent congruent rectangles are joined by an infinite line.

Original entry on oeis.org

0, 5, 17, 57, 133, 297, 525, 925, 1477, 2289, 3277, 4701, 6437, 8805, 11541, 14917, 18869, 23893, 29509, 36473, 44349, 53545, 63605, 75629, 88901, 104325, 120981, 139913, 160581, 184409, 209885, 238989, 270525, 305413, 342413, 383301, 426949, 475757, 527205, 583261, 642821, 708717, 777829

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Sep 12 2021

nonn

			a(1) = 5 as connecting the four vertices of a single rectangle forms one new vertex inside the rectangle, giving a total of 4 + 1 = 5 total intersection points.
a(2) = 17 as connecting the six vertices of two adjacent rectangles forms seven vertices inside the rectangles while also forming four vertices outside the rectangles. The total number of intersection points is then 6 + 7 + 4 = 17.
See the linked images for further examples.

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.

Cf. A344993 (number of polygons), A347751 (number of edges), A159065, A331755, A092275 (number of intersections resp. inside the rectangles, on or inside them, above them).

a(n) = A347751(n) - A344993(n) + 1.

It seems that a(n) = 2 * A159065(n+1) + 3 for n>0. - Andrei Zabolotskii, Jul 03 2025

A333277 a(n) = Sum_k k*A333275(n,k).

Original entry on oeis.org

4, 30, 116, 290, 652, 1186, 2092, 3370, 5212, 7546, 10828, 14866, 20260, 26674, 34508, 43778, 55364, 68546, 84644, 102962, 124172, 147842, 175772, 206738, 242372, 281506, 325652, 374090, 429356, 488938, 556348, 629738, 710468, 797210, 892492, 994514, 1107668

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 15 2020

nonn

a(n)/A331755(n) is the average number of polygons touching a non-boundary vertex in the graph defined in A306302.

Lars Blomberg, Table of n, a(n) for n = 1..100

Cf. A159065, A306302, A333274, A333275, A333276.

a(6) and beyond from Lars Blomberg, Jun 17 2020

A359691 Number of crossings in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.

Original entry on oeis.org

1, 7, 59, 275, 1949, 3971, 20333, 45705, 120899, 205233, 629761, 897707, 2334291, 3461329, 5516985, 8467899

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 11 2023

The number of vertices along each edge is A005728(n). No formula for a(n) is known.

See A359690 for images of the graph.

Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Wikipedia, Farey sequence.

Cf. A359690 (vertices), A359692 (regions), A359693 (edges), A359694 (k-gons), A005728, A159065, A331755, A359654, A358887, A358883, A006842, A006843.

a(n) = A359690(n) - 2*A005728(n).

A369176 Number of vertices in a graph of n adjacent rectangles in a row with all possible diagonals drawn, as in A306302, but without the rectangles' edges which are perpendicular to the row.

Original entry on oeis.org

5, 13, 31, 67, 139, 247, 429, 691, 1067, 1543, 2217, 3047, 4169, 5495, 7117, 9031, 11449, 14179, 17547, 21379, 25835, 30755, 36613, 43091, 50605, 58775, 68035, 78171, 89831, 102335, 116593, 132079, 149181, 167391, 187497, 208983, 232977, 258351, 285957, 315323, 347777, 381867, 419371

Offset: 1

Scott R. Shannon, Jan 15 2024

nonn

Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 8.

Cf. A369175 (regions), A369177 (edges), A369178 (k-gons), A306302, A331755, A368756.

a(n) = A369177(n) - A369175(n) + 1 by Euler's formula.

cp's OEIS Frontend

A333280 Column 2 of triangle in A333278.

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A333281 Column 2 of triangle in A288180.

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A333276 a(n) = Sum_k k*A333274(n,k).

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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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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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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.

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A347750 Number of intersection points when every pair of vertices of a row of n adjacent congruent rectangles are joined by an infinite line.

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Formula

A333277 a(n) = Sum_k k*A333275(n,k).

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Extensions

A359691 Number of crossings in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.

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A369176 Number of vertices in a graph of n adjacent rectangles in a row with all possible diagonals drawn, as in A306302, but without the rectangles' edges which are perpendicular to the row.

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