A331766 -id:A331766 - cp's OEIS frontend

A332609 Maximum number of edges in any cell in the graph formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) rectangular grid of points (or equally, a 2 X n grid of squares).

4, 4, 5, 5, 5, 6, 5, 6, 8, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 12 2020

nonn

See A331452 (the illustrations for T(n,2)) for pictures of these graphs.

Scott R. Shannon, Data specifically for nX2 (or 2Xn) rectangles

Cf. A331452, A331453, A331454, A331763, A331765, A331766, A332599, A332600, A331457, A332606, A332607, A332608.

a(21)-a(87) from Lars Blomberg, Apr 28 2020

A333278 Triangle read by rows: T(n,m) (n >= m >= 1) = number of edges in the graph formed by drawing the line segments connecting any two of the (n+1) X (m+1) lattice points in an n X m grid of squares.

Original entry on oeis.org

8, 28, 92, 80, 296, 872, 178, 652, 1922, 4344, 372, 1408, 4256, 9738, 21284, 654, 2470, 7466, 16978, 36922, 64172, 1124, 4312, 13112, 29874, 64800, 113494, 200028, 1782, 6774, 20812, 47402, 103116, 181484, 319516, 509584, 2724, 10428, 31776, 72398, 158352, 279070, 490396, 782096, 1199428

Offset: 1

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

T(n,m) = A288180(n,m)+A288187(n,m)-1 (Euler).

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.

			Triangle begins:
8,
28, 92,
80, 296, 872,
178, 652, 1922, 4344,
372, 1408, 4256, 9738, 21284,
654, 2470, 7466, 16978, 36922, 64172,
...

Hugo Pfoertner, Illustration of intersection points up to 6 X 6.

Cf. A288180.
For column 1 see A331757. For column 2 see A333279, A333280, A333281.
Cf. A331452, A288187; A331766, A331765, A331763; A333279, A333280, A333281.

A331767 Number of interior vertices formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) square grid.

Original entry on oeis.org

7, 29, 89, 201, 387, 641, 1067, 1599, 2305, 3233, 4431, 5855, 7721, 9853, 12369, 15477, 19093, 23141, 28073, 33557, 39699, 46773, 54815, 63681, 73825, 84833, 97005, 110579, 125587, 141733, 160063, 179913, 201105, 224409, 249329, 276215, 305925, 337345, 370793

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 05 2020

nonn

Same as A331763, but only count interior crossings.

See A331452 for other illustrations.

Lars Blomberg, Table of n, a(n) for n = 1..100
Scott R. Shannon, Colored illustration for a(3) = 89.

Cf. A331763, A331765, A331766.

Needs to be extended using the new data for A331763. - N. J. A. Sloane, Mar 12 2020

a(10) and beyond from Lars Blomberg, Apr 28 2020

A335701 Irregular triangle read by rows: consider the structure formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of an (n+1) X 3 rectangular grid of points (or equally, an n X 2 grid of squares); row n gives number of cells with k sides, for k >= 3.

Original entry on oeis.org

14, 2, 48, 8, 102, 36, 4, 192, 92, 12, 326, 194, 24, 524, 336, 28, 4, 802, 554, 80, 1192, 812, 128, 4, 1634, 1314, 112, 0, 4, 2, 2296, 1756, 200, 20, 3074, 2508, 236, 22, 4052, 3252, 356, 28, 5246, 4348, 472, 28, 6740, 5464, 652, 28, 8398, 7054, 656, 74, 10440, 8760, 940, 52, 12770, 11050, 1040, 58, 15512, 13324, 1300, 60, 4, 18782, 16162, 1600, 70, 22384, 19256, 1948, 104

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Aug 09 2020

More than the usual number of terms are given, in order to include the first 20 rows and emphasize the fact that so far k is never more than 8.
These are the structures discussed in column 2 of the table in A331452. It is known that the structures discussed in column 1 of that table have cells with at most 4 sides, so an upper limit of 8 sides for the present sequence is certainly possible.
The maximum number of sides for n=19..106 is 6. - Lars Blomberg, Aug 27 2020

			Triangle begins:
14, 2,
48, 8,
102, 36, 4,
192, 92, 12
326, 194, 24
524, 336, 28, 4
802, 554, 80,
1192, 812, 128, 4
1634, 1314, 112, 0, 4, 2
2296, 1756, 200, 20
3074, 2508, 236, 22
4052, 3252, 356, 28
5246, 4348, 472, 28
6740, 5464, 652, 28
8398, 7054, 656, 74
10440, 8760, 940, 52
12770, 11050, 1040, 58
15512, 13324, 1300, 60, 4
18782, 16162, 1600, 70
22384, 19256, 1948, 104
...
The 1X2 structure (or 2X1 structure, as in the illustration) contains 14 triangles and 2 quadrilaterals, so row 1 is 14, 2.
The 3X2 structure contains 102 triangles, 36 quadrilaterals, and 4 pentagons, so row 3 is 102, 36, 4. The sum is 142 = A331766(3).

Lars Blomberg, Table of n, a(n) for n = 1..419 (the first 106 rows)
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 for n=1
Scott R. Shannon, Colored illustration for n=2
Scott R. Shannon, Colored illustration for n=3
Scott R. Shannon, Colored illustration for n=4
Scott R. Shannon, Colored illustration for n=5
Scott R. Shannon, Colored illustration for n=6
Scott R. Shannon, Colored illustration for n=7
Scott R. Shannon, Colored illustration for n=8
Scott R. Shannon, Colored illustration for n=9
Scott R. Shannon, Colored illustration for n=10
Scott R. Shannon, Colored illustration for n=11
Scott R. Shannon, Colored illustration for n=12
Scott R. Shannon, Colored illustration for n=13
Scott R. Shannon, Colored illustration for n=14
Scott R. Shannon, Colored illustration for n=15
Scott R. Shannon, Colored illustration for n=16
Scott R. Shannon, Colored illustration for n=2. Random distance-based coloring.
Scott R. Shannon, Colored illustration for n=3. Random distance-based coloring.
Scott R. Shannon, Colored illustration for n=4. Random distance-based coloring.
Scott R. Shannon, Colored illustration for n=5. Random distance-based coloring.
Scott R. Shannon, Colored illustration for n=6. Random distance-based coloring.
Scott R. Shannon, Colored illustration for n=7. Random distance-based coloring.
Scott R. Shannon, Colored illustration for n=8. Random distance-based coloring.
Scott R. Shannon, Colored illustration for n=9. Random distance-based coloring.
Scott R. Shannon, Colored illustration for n=10. Random distance-based coloring.

Cf. A331452, A331766 (row sums), A331763, A331765.

A337229 Irregular table read by rows: row n gives the number of 3-gon to k-gon contacts, with k>=3, for an n X 2 grid of congruent rectangles divided by drawing diagonals of all possible rectangles (cf. A335701).

Original entry on oeis.org

14, 8, 56, 24, 88, 104, 16, 156, 204, 48, 208, 460, 88, 328, 760, 116, 24, 498, 1088, 304, 716, 1608, 492, 24, 894, 2632, 420, 0, 24, 20, 1282, 3436, 756, 108, 1730, 4780, 852, 104, 2156, 6332, 1348, 136, 2844, 8236, 1632, 152, 3594, 10464, 2388, 148, 4432, 13632, 2260, 404

Offset: 1

Scott R. Shannon, Aug 20 2020

See A335701 for other images of the n X 2 grid of congruent rectangles.

			The table begins:
.
14, 8;
56, 24;
88, 104, 16;
156, 204, 48;
208, 460, 88;
328, 760, 116, 24;
498, 1088, 304;
716, 1608, 492, 24;
894, 2632, 420, 0, 24, 20;
1282, 3436, 756, 108;
1730, 4780, 852, 104;
2156, 6332, 1348, 136;
2844, 8236, 1632, 152;
3594, 10464, 2388, 148;
4432, 13632, 2260, 404;
5530, 16604, 3354, 256;
6812, 20700, 3668, 280;

Scott R. Shannon, Image for n=1 with edge-count coloring. This is the first n-gon with 3-gon to 3-gon and 3-gon to 4-gon contacts.
Scott R. Shannon, Image for n=3 with edge-count coloring. This is the first n-gon with 3-gon to 5-gon contacts.
Scott R. Shannon, Image for n=6 with edge-count coloring. This is the first n-gon with 3-gon to 6-gon contacts.
Scott R. Shannon, Image for n=9 with edge-count coloring. This is the first n-gon with 3-gon to 7-gon and 3-gon to 8-gon contacts.

Cf. A335701, A331454, A331766, A331763, A336731.

cp's OEIS Frontend

A332609 Maximum number of edges in any cell in the graph formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) rectangular grid of points (or equally, a 2 X n grid of squares).

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A333278 Triangle read by rows: T(n,m) (n >= m >= 1) = number of edges in the graph formed by drawing the line segments connecting any two of the (n+1) X (m+1) lattice points in an n X m grid of squares.

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A331767 Number of interior vertices formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) square grid.

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A335701 Irregular triangle read by rows: consider the structure formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of an (n+1) X 3 rectangular grid of points (or equally, an n X 2 grid of squares); row n gives number of cells with k sides, for k >= 3.

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A337229 Irregular table read by rows: row n gives the number of 3-gon to k-gon contacts, with k>=3, for an n X 2 grid of congruent rectangles divided by drawing diagonals of all possible rectangles (cf. A335701).

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