A306302 -id:A306302 - cp's OEIS frontend

A333288 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all visible rectangles; T(n,k) (1 <= k <= n) is the number of regions in the k-th rectangle.

4, 8, 8, 12, 22, 12, 16, 36, 36, 16, 20, 52, 70, 52, 20, 24, 66, 100, 100, 66, 24, 28, 82, 134, 160, 134, 82, 28, 32, 98, 166, 218, 218, 166, 98, 32, 36, 116, 198, 276, 310, 276, 198, 116, 36, 40, 134, 230, 328, 396, 396, 328, 230, 134, 40, 44, 154, 266, 386

Offset: 1

N. J. A. Sloane, Mar 20 2020

This was originally based on the data in Jinyuan Wang's A324042, and then extended by Lars Blomberg.
Since the cells are either triangles or quadrilaterals, this is the sum of the two arrays A333286 and A333287.
It would be nice to have a formula for these entries. It is easy to see that the first column is 4n for n>=1.

			Triangle begins:
   4;
   8,   8;
  12,  22,  12;
  16,  36,  36,  16;
  20,  52,  70,  52,  20;
  24,  66, 100, 100,  66,  24;
  28,  82, 134, 160, 134,  82,  28;
  ...

Lars Blomberg, Table of n, a(n) for n = 1..3240 (the first 80 rows)
Lars Blomberg, Scott R. Shannon and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.

Cf. A306302, A331452, A324042, A324043, A333286, A333287, A333288, A335056, A335074.

a(29) and beyond from Lars Blomberg, Apr 23 2020

A324042 Number of triangular regions into which a figure made up of a row of n adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles.

Original entry on oeis.org

4, 14, 32, 70, 124, 226, 360, 566, 820, 1218, 1696, 2310, 3020, 4018, 5160, 6590, 8196, 10218, 12464, 15110, 18012, 21650, 25624, 30142, 35028, 40954, 47344, 54558, 62284, 71034, 80360, 90806, 101892, 114770, 128416, 143286, 158972, 176914, 195816, 216350, 237908, 261546, 286304, 313102, 341100

Offset: 1

Jinyuan Wang, May 01 2019

nonn

A row of n adjacent congruent rectangles can only be divided into triangles or quadrilaterals when drawing diagonals. A proof is given in Alekseyev et al. (2015) using the mapping to a dissection of a a right isosceles triangle described in A306302.

			For k adjacent congruent rectangles, the number of triangular regions in the j-th rectangle is:
k\j|  1   2   3   4   5   6   7  ...
---+--------------------------------
1  |  4,  0,  0,  0,  0,  0,  0, ...
2  |  7,  7,  0,  0,  0,  0,  0, ...
3  |  9, 14,  9,  0,  0,  0,  0, ...
4  | 11, 24, 24, 11,  0,  0,  0, ...
5  | 13, 30, 38, 30, 13,  0,  0, ...
6  | 15, 38, 60, 60, 38, 15,  0, ...
7  | 17, 44, 76, 86, 76, 44, 17, ...
...
a(4) = 11 + 24 + 24 + 11 = 70.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
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.
Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.
Robert Israel, Maple program
Jinyuan Wang, Illustration for n = 1, 2, 3, 4, 5

Cf. A177719, A306302, A324043, A333286, A333287, A333288.

V := proc(m,n,q) local a,i,j; a:=0;
for i from 1 to m do for j from 1 to n do
if gcd(i,j)=q then a:=a+(m+1-i)*(n+1-j); fi; od: od: a; end;
a := n -> 2*( n*(n+1) + V(n,n,2) );
[seq(a(n), n=1..30)]; # N. J. A. Sloane, Mar 04 2020
See also Robert Israel link.

Table[2 * (n^2 + n + Sum[Sum[Boole[GCD[i, j] == 2] * (n + 1 - i) * (n + 1 - j), {j, 1, n}], {i, 1, n}]), {n, 1, 45}]  (* Joshua Oliver, Feb 05 2020 *)

{ A324042(n) = 2*((n+1)*n + sum(i=1, n, sum(j=1, n, (gcd(i, j)==2)*(n+1-i)*(n+1-j))) ); } \\ Max Alekseyev, Jul 08 2019

from sympy import totient
def A324042(n): return 2*(2*n**2-n+1 + 2*sum(totient(i)*(n+1-2*i)*(n+1-i) for i in range(2,n//2+1))) # Chai Wah Wu, Aug 16 2021

a(n) = A177719(n+1) + 2*(n+1) = 2 * ( (n+1)*n + Sum_{i,j=1..n; gcd(i,j)=2} (n+1-i)*(n+1-j) ). - Max Alekseyev, Jul 08 2019
a(n) = A306302(n) - A324043(n).
a(n) = 2*(2*n^2-n+1+2*Sum_{i=2..floor(n/2)} (n+1-2*i)*(n+1-i)*phi(i)). - Chai Wah Wu, Aug 16 2021

a(8)-a(23) from Robert Israel, Jul 07 2019

Terms a(24) onward from Max Alekseyev, Jul 08 2019

A331457 Triangle read by rows: T(n,k) = number of regions in a "frame" of size n X k (see Comments for definition).

Original entry on oeis.org

4, 16, 56, 46, 142, 208, 104, 296, 348, 496, 214, 544, 592, 752, 1016, 380, 892, 948, 1120, 1396, 1784, 648, 1436, 1508, 1692, 1980, 2380, 2984, 1028, 2136, 2292, 2488, 2788, 3200, 3816, 4656, 1562, 3066, 3384, 3592, 3904, 4328, 4956, 5808, 6968, 2256, 4272, 4796, 5016, 5340, 5776, 6416, 7280, 8452, 9944

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 29 2020

A "frame" of size n X k is formed from a grid of (n+1) X (k+1) points with the central grid of (n-3) X (k-3) points removed. If n or k is less than 3 then no points are removed, and T(n,k) = A331452(n,k). From now on we assume both n and k are >= 3.
The resulting array has an outer perimeter with 2*(n+k) points and an inner perimeter with 2*(n+k)-8 points, for a total of 4*(n+k)-8 perimeter points. The frame itself is the strip of width 1 between the inner and outer perimeters.
Now join every pair of perimeter points, both inner and outer, by a line segment, provided the line remains inside the frame. The sequence gives the number of regions in the resulting figure.
See A331776 for additional illustrations for the diagonal entries.
There is a crucial difference between frames of size nX2 and size nXk with k = 1 or k >= 3. If k != 2, all regions are either triangles or quadrilaterals, but for k=2 regions with larger numbers of sides can appear. Remember also that for k <= 2, the "frame" has no hole, and the graph has genus 0, whereas for k >= 3 there is a nontrivial hole and the graph has genus 1.

			Triangle begins:
4,
16,56,
46,142,208,
104,296,348,496,
214,544,592,752,1016
380,892,948,1120,1396,1784
648,1436,1508,1692,1980,2380,2984
1028,2136,2292,2488,2788,3200,3816,4656
1562,3066,3384,3592,3904,4328,4956,5808,6968
2256,4272,4796,5016,5340,5776,6416,7280,8452,9944

Scott R. Shannon, Colored illustration for T(1,1) = 4.
Scott R. Shannon, Colored illustration for T(2,2) = 56.
Scott R. Shannon, Colored illustration for T(3,3) = 208.
Scott R. Shannon, Colored illustration for T(4,4) = 496.
Scott R. Shannon, Colored illustration for T(5,5) = 1016.
Scott R. Shannon, Colored illustration for T(6,6) = 1784.
Scott R. Shannon, Colored illustration for T(7,4) = 1692.
Scott R. Shannon, Colored illustration for T(10,6) = 5776.
N. J. A. Sloane, Illustration for T(3,3) = 208.

Cf. A332599 (triangle giving numbers of vertices) and A332600 (edges).
Cf. also A331452.
The first column is A306302, the main diagonal is A331776.

Column 1 is A306302, for which there is an explicit formula.
Column 2 is A331766, for which no formula is known.
For n >= k >= 3, T(n,k) = A332610(n,k) + A332611(n,k), both of which have explicit formulas.

More terms from Scott R. Shannon, Mar 05 2020

a(8) corrected by Giovanni Resta, May 22 2025

A344993 Number of polygons formed 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, 4, 20, 68, 168, 368, 676, 1184, 1912, 2944, 4292, 6152, 8456, 11484, 15164, 19624, 24944, 31508, 39076, 48212, 58656, 70672, 84284, 100192, 117888, 138100, 160580, 185796, 213568, 245008, 279116, 317424, 359280, 405124, 454868, 509264, 567640, 631988, 701228, 776032, 855968, 943260, 1035844

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Jun 05 2021

nonn

The number of polygons formed inside the rectangles is A306302(n), while the number of polygons formed outside the rectangles is 2*A332612(n+1).
The number of open regions, those outside the polygons with unbounded area and two edges that go to infinity, for n >= 1 is given by 2*n^2 + 4*n + 6 = A255843(n+1).
Like A306302(n) is appears only 3-gons and 4-gons are generated by the infinite lines.

			a(1) = 4 as connecting the four vertices of a single rectangle forms four triangles inside the rectangle. Twelve open regions outside these triangles are also formed.
a(2) = 20 as connecting the six vertices of two adjacent rectangles forms two quadrilaterals and fourteen triangles inside the rectangles while also forming four triangles outside the rectangles, giving twenty polygons in total. Twenty-two open regions outside these polygons are also formed.
See the linked images for further examples.

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Image for n = 1. In this and other images the vertices at the corners of all rectangles are highlighted as white dots while the outer open regions, which are not counted, are darkened.
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.

See A347750 and A347751 for the numbers of vertices and edges in the finite part of the corresponding graph.

Cf. A332612 (half the number of polygons outside the rectangles), A306302 (number of polygons inside the rectangles), A255843.

from sympy import totient
def A344993(n): return 2*n*(n+1) + 2*sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(2,n+1)) # Chai Wah Wu, Aug 21 2021

a(n) = 2*A332612(n+1) + A306302(n) = 2*Sum_{i=2..n, j=1..i-1, gcd(i,j)=1} (n+1-i)*(n+1-j) + Sum_{i=1..n, j=1..n, gcd(i,j)=1} (n+1-i)*(n+1-j) + n^2 + 2*n.

a(n) = 2*n*(n+1) + 2*Sum_{i=2..n} (n+1-i)*(2*n+2-i)*phi(i). - Chai Wah Wu, Aug 21 2021

A333286 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all visible rectangles; T(n,k) (1 <= k <= n) is the number of triangular regions in the k-th rectangle.

Original entry on oeis.org

4, 7, 7, 9, 14, 9, 11, 24, 24, 11, 13, 30, 38, 30, 13, 15, 38, 60, 60, 38, 15, 17, 44, 76, 86, 76, 44, 17, 19, 52, 92, 120, 120, 92, 52, 19, 21, 58, 106, 146, 158, 146, 106, 58, 21, 23, 66, 126, 178, 216, 216, 178, 126, 66, 23, 25, 72, 142, 206, 264, 278

Offset: 1

N. J. A. Sloane, Mar 20 2020

This was originally based on the data in Jinyuan Wang's A324042, and then extended by Lars Blomberg.

It would be nice to have a formula for these entries. It is easy to see that the first column is 2n+3 for n>1.

			Triangle begins:
4,
7,  7,
9, 14,  9,
11, 24, 24, 11,
13, 30, 38, 30, 13,
15, 38, 60, 60, 38, 15,
17, 44, 76, 86, 76, 44, 17,
...

Lars Blomberg, Table of n, a(n) for n = 1..3240 (the first 80 rows)
Lars Blomberg, Scott R. Shannon and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.

Cf. A306302, A331452, A324042, A324043, A333287, A333288.

a(29) and beyond from Lars Blomberg, Apr 23 2020

A333287 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all visible rectangles; T(n,k) (1 <= k <= n) is the number of quadrilateral regions in the k-th rectangle.

Original entry on oeis.org

0, 1, 1, 3, 8, 3, 5, 12, 12, 5, 7, 22, 32, 22, 7, 9, 28, 40, 40, 28, 9, 11, 38, 58, 74, 58, 38, 11, 13, 46, 74, 98, 98, 74, 46, 13, 15, 58, 92, 130, 152, 130, 92, 58, 15, 17, 68, 104, 150, 180, 180, 150, 104, 68, 17, 19, 82, 124, 180, 224, 254, 224, 180, 124, 82, 19

Offset: 1

N. J. A. Sloane, Mar 20 2020

This was originally based on the data in Jinyuan Wang's A324042, and then extended by Lars Blomberg.

It would be nice to have a formula for these entries. It is easy to see that the first column is 2n-3 for n>1.

			Triangle begins:
0,
1,  1,
3,  8,  3,
5, 12, 12,  5,
7, 22, 32, 22,  7,
9, 28, 40, 40, 28,  9,
11, 38, 58, 74, 58, 38, 11,
...

Lars Blomberg, Table of n, a(n) for n = 1..3240 (the first 80 rows)
Lars Blomberg, Scott R. Shannon and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.

Cf. A306302, A331452, A324042, A324043, A333286, A333288.

a(29) and beyond from Lars Blomberg, Apr 23 2020

A334701 Consider the figure made up of a row of n adjacent congruent rectangles, with diagonals of all possible rectangles drawn; a(n) = number of interior vertices where exactly two lines cross.

Original entry on oeis.org

1, 6, 24, 54, 124, 214, 382, 598, 950, 1334, 1912, 2622, 3624, 4690, 6096, 7686, 9764, 12010, 14866, 18026, 21904, 25918, 30818, 36246, 42654, 49246, 57006, 65334, 75098, 85414, 97384, 110138, 124726, 139642, 156286, 174018, 194106, 214570, 237534, 261666, 288686, 316770, 348048, 380798, 416524, 452794, 492830

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 30 2020

nonn

It would be nice to have a formula or recurrence. - N. J. A. Sloane, Jun 22 2020

Lars Blomberg, Table of n, a(n) for n = 1..500
Lars Blomberg, Array (s,n) of the number of internal vertices where exactly s=2..501 lines cross in a figure made up of a row of n=1..500 adjacent congruent rectangles, with diagonals of all possible rectangles drawn. Rows are stored comma-separated.
Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
Scott R. Shannon, Colored illustration showing regions for n=1
Scott R. Shannon, Images of vertices for n=1.
Scott R. Shannon, Colored illustration showing regions for n=2
Scott R. Shannon, Images of vertices for n=2.
Scott R. Shannon, Colored illustration showing regions for n=3
Scott R. Shannon, Images of vertices for n=3.
Scott R. Shannon, Colored illustration showing regions for n=4
Scott R. Shannon, Images of vertices for n=4.
Scott R. Shannon, Colored illustration showing regions for n=5
Scott R. Shannon, Images of vertices for n=5
Scott R. Shannon, Colored illustration showing regions for n=6
Scott R. Shannon, Images of vertices for n=6
Scott R. Shannon, Images of vertices for n=7
Scott R. Shannon, Images of vertices for n=8
Scott R. Shannon, Images of vertices for n=9.
Scott R. Shannon, Images of vertices for n=11.
Scott R. Shannon, Images of vertices for n=14.
Index entries for sequences related to stained glass windows

Column 4 of array in A333275.
Cf. A306302, A331755, A290131, A333274.
See also A115004, A331761.

Conjecture: As n -> oo, a(n) ~ C*n^4/Pi^2, where C is about 0.95 (compare A115004, A331761). - N. J. A. Sloane, Jul 03 2020

More terms from Lars Blomberg, Jun 17 2020

A332358 Main diagonal of A332357.

Original entry on oeis.org

1, 5, 17, 47, 105, 215, 381, 649, 1029, 1563, 2257, 3209, 4385, 5925, 7793, 10053, 12745, 16061, 19881, 24487, 29749, 35799, 42649, 50649, 59545, 69701, 80993, 93655, 107597, 123375, 140489, 159705, 180697, 203685, 228625, 255893, 285153, 317401, 352097, 389577

Offset: 1

N. J. A. Sloane, Feb 11 2020

nonn

Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.
Scott R. Shannon, Illustration for n = 4.
Scott R. Shannon, Illustration for n = 5.
Scott R. Shannon, Illustration for n = 10.
Scott R. Shannon, Illustration for n = 11.

Cf. A332357, A306302.

a(n) = A306302(n-1) + 1. [see Max Alekseyev's comment in A306302] - Andrey Zabolotskiy, Sep 14 2023

A333279 Column 2 of triangle in A288187.

Original entry on oeis.org

16, 56, 176, 388, 822, 1452, 2516, 3952, 6060, 8736, 12492, 17040, 23102, 30280, 39234, 49688, 62730, 77556, 95642, 115992, 139874, 166560, 197992, 232600, 272574, 316460, 366390, 420792, 482748, 549516, 624962, 706436, 796766, 893844, 1001074, 1115428

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.

The maximum number of edges over all chambers is 4 for 1 <= n <= 4 and 5 for 5 <= n <= 160. - Lars Blomberg, May 23 2021

Lars Blomberg, Table of n, a(n) for n = 1..200
Lars Blomberg, Colored illustration of a(1)
Lars Blomberg, Colored illustration of a(2)
Lars Blomberg, Colored illustration of a(3)
Lars Blomberg, Colored illustration of a(4)
Lars Blomberg, Colored illustration of a(5)
Lars Blomberg, Colored illustration of a(6)
Lars Blomberg, Colored illustration of a(7)
Lars Blomberg, Colored illustration of a(8)
Lars Blomberg, Colored illustration of a(9)
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

A333280 Column 2 of triangle in A333278.

Original entry on oeis.org

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

cp's OEIS Frontend

A333288 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all visible rectangles; T(n,k) (1 <= k <= n) is the number of regions in the k-th rectangle.

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A324042 Number of triangular regions into which a figure made up of a row of n adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles.

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A331457 Triangle read by rows: T(n,k) = number of regions in a "frame" of size n X k (see Comments for definition).

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A344993 Number of polygons formed when every pair of vertices of a row of n adjacent congruent rectangles are joined by an infinite line.

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A333286 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all visible rectangles; T(n,k) (1 <= k <= n) is the number of triangular regions in the k-th rectangle.

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A333287 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all visible rectangles; T(n,k) (1 <= k <= n) is the number of quadrilateral regions in the k-th rectangle.

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A334701 Consider the figure made up of a row of n adjacent congruent rectangles, with diagonals of all possible rectangles drawn; a(n) = number of interior vertices where exactly two lines cross.

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A332358 Main diagonal of A332357.

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