A331755 -id:A331755 - cp's OEIS frontend

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.

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.

A333284 Triangle read by rows: T(m,n) (m >= n >= 1) = number of vertices formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.

Original entry on oeis.org

5, 13, 37, 35, 129, 405, 75, 289, 933, 2225, 159, 663, 2155, 5157, 11641, 275, 1163, 3793, 9051, 20341, 35677, 477, 2069, 6771, 16129, 36173, 63987, 114409, 755, 3251, 10727, 25635, 57759, 102845, 183961

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 16 2020

If we only joined pairs of the 2(m+n) boundary points, we would get A331453. If we did not extend the lines to the boundary of the grid, we would get A288180. (One of the links below shows the difference between the three definitions in the case m=3, n=2.)

See A333282 for a large number of colored illustrations.

			Triangle begins:
5,
13, 37,
35, 129, 405,
75, 289, 933, 2225,
159, 663, 2155, 5157, 11641,
275, 1163, 3793, 9051, 20341, 35677,
477, 2069, 6771, 16129, 36173, 63987, 114409,
755, 3251, 10727, 25635, 57759, 102845, 183961, ...
...
T(7,7) corrected Mar 19 2020

Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.
Seppo Mustonen, Statistical accuracy of geometric constructions, 2008.
Seppo Mustonen, Statistical accuracy of geometric constructions, 2008 [Local copy]
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009 [Local copy]
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010 [Local copy]
N. J. A. Sloane, Illustration of T(3,2) = 129. [Black lines correspond to A331453(3,2), black + red lines correspond to A288180(3,2), and black + red + blue lines to T(3,2)]
N. J. A. Sloane, Illustration of T(3,3) = 405 [Black lines correspond to A288180(3,3), and black + red lines to T(3,3)]

Cf. A288187, A331452, A288180, A331453, A333282 (regions), A333283 (edges). Column 1 is A331755. The main diagonal is A333285.

More terms and corrections from Scott R. Shannon, Mar 21 2020

A335056 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all possible rectangles; T(n,k) (1 <= k <= n) is the number of vertices inside the k-th rectangle.

Original entry on oeis.org

1, 3, 3, 5, 11, 5, 7, 19, 19, 7, 9, 29, 43, 29, 9, 11, 37, 61, 61, 37, 11, 13, 47, 83, 105, 83, 47, 13, 15, 57, 103, 143, 143, 103, 57, 15, 17, 69, 125, 183, 211, 183, 125, 69, 17, 19, 81, 143, 215, 267, 267, 215, 143, 81, 19, 21, 95, 167, 253, 329, 369, 329, 253, 167, 95, 21, 23, 109, 189, 289, 385, 455, 455, 385, 289, 189, 109, 23

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 21 2020

The terms are from numeric computation - no formula for a(n) is currently known.

			Triangle begins:
1;
3, 3;
5, 11, 5;
7, 19, 19, 7;
9, 29, 43, 29, 9;
11, 37, 61, 61, 37, 11;
13, 47, 83, 105, 83, 47, 13;
15, 57, 103, 143, 143, 103, 57, 15;
17, 69, 125, 183, 211, 183, 125, 69, 17;
19, 81, 143, 215, 267, 267, 215, 143, 81, 19;
21, 95, 167, 253, 329, 369, 329, 253, 167, 95, 21;
23, 109, 189, 289, 385, 455, 455, 385, 289, 189, 109, 23;
25, 125, 215, 331, 451, 551, 597, 551, 451, 331, 215, 125, 25;

Scott R. Shannon, Image for n = 2 showing the count of the vertices.
Scott R. Shannon, Image for n = 3 showing the count of the vertices.
Scott R. Shannon, Image for n = 5 showing the count of the vertices.
Scott R. Shannon, Image for n = 9 showing the count of the vertices.
Scott R. Shannon, Image for n = 12 showing the count of the vertices.

Cf. A335074, A159065, A331755, A333288, A306302.

Row sum n + Row sum A335074(n) = A159065(n).

A335074 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all possible rectangles; T(n,k) (1 <= k <= n-1) is the number of vertices on the edge separating rectangles k and k+1.

Original entry on oeis.org

1, 3, 3, 5, 3, 5, 7, 7, 7, 7, 9, 9, 7, 9, 9, 11, 13, 11, 11, 13, 11, 13, 15, 17, 11, 17, 15, 13, 15, 19, 19, 19, 19, 19, 19, 15, 17, 21, 25, 21, 19, 21, 25, 21, 17, 19, 25, 29, 29, 23, 23, 29, 29, 25, 19, 21, 27, 33, 33, 33, 23, 33, 33, 33, 27, 21, 23, 31, 37, 39, 39, 35, 35, 39, 39, 37, 31, 23

Offset: 2

Scott R. Shannon and N. J. A. Sloane, May 22 2020

The terms are from numeric computation - no formula for a(n) is currently known.

			Triangle begins:
1;
3, 3;
5, 3, 5;
7, 7, 7, 7;
9, 9, 7, 9, 9;
11, 13, 11, 11, 13, 11;
13, 15, 17, 11, 17, 15, 13;
15, 19, 19, 19, 19, 19, 19, 15;
17, 21, 25, 21, 19, 21, 25, 21, 17;
19, 25, 29, 29, 23, 23, 29, 29, 25, 19;
21, 27, 33, 33, 33, 23, 33, 33, 33, 27, 21;
23, 31, 37, 39, 39, 35, 35, 39, 39, 37, 31, 23;

Scott R. Shannon, Image for n = 3 showing the count of the vertices.
Scott R. Shannon, Image for n = 4 showing the count of the vertices.
Scott R. Shannon, Image for n = 7 showing the count of the vertices.
Scott R. Shannon, Image for n = 10 showing the count of the vertices.
Scott R. Shannon, Image for n = 12 showing the count of the vertices.

Cf. A335056, A159065, A331755, A333288, A306302.

Row sum n + Row sum A335056(n) = A159065(n).

A336731 Three-column table read by rows: row n gives [number of triangle-triangle, triangle-quadrilateral, quadrilateral-quadrilateral] contacts for a row of n adjacent congruent rectangles divided by drawing diagonals of all possible rectangles (cf. A331452).

Original entry on oeis.org

4, 0, 0, 14, 8, 0, 20, 48, 4, 60, 80, 28, 68, 224, 68, 148, 368, 124, 224, 616, 268, 336, 1008, 420, 384, 1672, 648, 712, 2208, 972, 972, 3120, 1464, 1300, 4304, 1996, 1496, 6040, 2788, 2044, 7936, 3580, 2612, 10224, 4672, 3540, 12656, 5980, 4224, 16104, 7676, 5484, 19648, 9500

Offset: 1

Scott R. Shannon, Aug 02 2020

For a row of n adjacent rectangles the only polygons formed when dividing all possible rectangles along their diagonals are 3-gons (triangles) and 4-gons (quadrilaterals). Hence the only possible edge-sharing contacts are 3-gons with 3-gons, 3-gons with 4-gons, and 4-gons with 4-gons. This sequence lists the number of these three possible combinations for a row of n adjacent rectangles. Note that the edges along the outside of the n adjacent rectangles are not counted as they are only in one n-gon.

These are graphs T(1,n) described in A331452. - N. J. A. Sloane, Aug 03 2020

			a(1) = 4, a(2) = 0, a(3) = 0. A single rectangle divided along its diagonals consists of four 3-gons, four edges, and no 4-gons. Therefore there are only four 3-gon-to-3-gon contacts. See the link image for n = 1.
a(4) = 14, a(5) = 8, a(6) = 0. Two adjacent rectangles divided along all diagonals consists of fourteen 3-gons and two 4-gons. The two 4-gons are separated and thus share all their edges, eight in total, with 3-gons. There are fourteen pairs of 3-gon-to-3-gon contacts. See the link image for n = 2.
a(7) = 20, a(8) = 48, a(9) = 4. Three adjacent rectangles divided along all diagonals consists of thirty-two 3-gons and fourteen 4-gons. There are two groups of three adjacent 4-gons, so there are four 4-gons-to-4-gon contacts. These, along with the other 4-gons, share 48 edges with 3-gons. There are also twenty 3-gon-to-3-gon contacts. See the link image for n = 3.
.
The table begins:
4,0,0;
14,8,0;
20,48,4;
60,80,28;
68,224,68;
148,368,124;
224,616,268;
336,1008,420;
384,1672,648;
712,2208,972;
972,3120,1464;
1300,4304,1996;
1496,6040,2788;
2044,7936,3580;
2612,10224,4672;
3540,12656,5980;
4224,16104,7676;
5484,19648,9500;
6568,24216,11936;
7836,29616,14468;
See A306302 for a count of the regions and images for other values of n.

Scott R. Shannon, Image of the rectangles for n = 1.
Scott R. Shannon, Image of the rectangles for n = 2.
Scott R. Shannon, Image of the rectangles for n = 3.
Scott R. Shannon, Image of the rectangles for n = 4.

Cf. A306302, A331452, A331755, A331757, A333288.

Sum of row t = A331757(t) - 2(t + 1).

A092275 Draw a line through every pair of points with coordinates (x, 1) and (x', 2) with x, x' in 1..n, and then count the number of intersection points above the line y = 2.

Original entry on oeis.org

0, 2, 11, 29, 69, 125, 224, 361, 563, 809, 1164, 1597, 2188, 2871, 3714, 4701, 5956, 7359, 9099, 11067, 13365, 15879, 18884, 22201, 26056, 30219, 34951, 40117, 46073, 52441, 59716, 67599, 76320, 85569, 95790, 106701, 118902

Offset: 2

N. J. A. Sloane, Feb 17 2004

nonn

Joseph O'Rourke, Art Gallery Theorems and Algorithms, Oxford Univ. Press, 1987. See section 8.6, p. 223.

Cf. A331755, A347750.

a(n) = (A347750(n-1) - A331755(n)) / 2. - Andrey Zabolotskiy, Jan 17 2024

New name and terms a(10) and beyond (using the formula) added by Andrey Zabolotskiy, Jan 18 2024

A334224 Consider a graph as defined in A306302 formed from a row of n adjacent congruent squares with the diagonals of all possible rectangles; a(n) is the minimum edge length of the squares such that the vertices formed by all intersections have integer x and y coordinates.

Original entry on oeis.org

2, 6, 60, 420, 2520, 27720, 360360, 360360, 12252240, 232792560, 232792560, 5354228880, 26771144400, 80313433200, 2329089562800, 72201776446800, 144403552893600, 144403552893600, 5342931457063200

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 28 2020

nonn

			a(1) = 2 as for a single square, with its bottom left corner at the origin, with both diagonals drawn the intersection point of those lines is at (L/2,L/2) where L is the edge length. Thus L=2 for this to have integer coordinates.
a(2) = 6 as for two vertically adjacent squares the seven intersection points of the diagonals and shared internal edge have coordinates (L/3,4L/3),(L/2,3L/2),(2L/3,4L/3),(L/2,L),(L/3,2L/3),(L/2,L/2),(2L/3,2L/3). Thus L=6, the lowest common multiple of the denominators, for all these points to have integer coordinates.

Cf. A306302, A003418, A331755, A290132, A290131.

a(n) = A003418(2n-1) = A076100(n) for n>1.

cp's OEIS Frontend

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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A333284 Triangle read by rows: T(m,n) (m >= n >= 1) = number of vertices formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.

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A335056 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all possible rectangles; T(n,k) (1 <= k <= n) is the number of vertices inside the k-th rectangle.

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A335074 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all possible rectangles; T(n,k) (1 <= k <= n-1) is the number of vertices on the edge separating rectangles k and k+1.

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A336731 Three-column table read by rows: row n gives [number of triangle-triangle, triangle-quadrilateral, quadrilateral-quadrilateral] contacts for a row of n adjacent congruent rectangles divided by drawing diagonals of all possible rectangles (cf. A331452).

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A092275 Draw a line through every pair of points with coordinates (x, 1) and (x', 2) with x, x' in 1..n, and then count the number of intersection points above the line y = 2.

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A334224 Consider a graph as defined in A306302 formed from a row of n adjacent congruent squares with the diagonals of all possible rectangles; a(n) is the minimum edge length of the squares such that the vertices formed by all intersections have integer x and y coordinates.

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