A331452 -id:A331452 - cp's OEIS frontend

A332953 The number of regions formed inside an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

1, 5, 18, 52, 125, 257, 486, 832, 1333, 2027, 3048, 4304, 6057, 8167, 10749, 13929, 18058, 22664, 28533, 34981, 42519, 51425, 62118, 73473, 86768, 101902, 118695, 137138, 159147, 181752, 208813, 237209, 268614, 303718, 340882, 380811, 427540, 477134, 530047

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 04 2020

nonn

The terms are from numeric computation - no formula for a(n) is currently known.
Equivalently, this is also the number of regions formed when all the integer points along the x and y axes with 0 <= x <= n and 0 <= y <= n are joined by straight line segments.
If instead one takes points on the x and y axes with coordinates 1, 1/2, 1/3, 1/4, ..., 1/n, 0, and joins them all by line segments, the resulting figure contains only triangles and quadrilaterals, and the number of regions is given by A332358 (and more generally by A332357 if there are m+1 such points on the x axis and n+1 such points on the y axis).

Lars Blomberg, Table of n, a(n) for n = 1..70
Scott R. Shannon, Illustration for n = 2.
Scott R. Shannon, Illustration for n = 3.
Scott R. Shannon, Illustration for n = 4.
Scott R. Shannon, Illustration for n = 5.
Scott R. Shannon, Illustration for n = 6.
Scott R. Shannon, Illustration for n = 8.
Scott R. Shannon, Illustration for n = 10.
Scott R. Shannon, Illustration for n = 12.
Scott R. Shannon, Illustration for n = 15.
Scott R. Shannon, Illustration for n = 5 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 10 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 15 with random distance-based coloring.

Cf. A333025 (n-gons), A333026 (vertices), A333027 (edges), A007678, A092867, A331452, A331911, A332357, A332358.

a(16) and beyond from Lars Blomberg, May 26 2020

A333025 Irregular table read by rows: Take an isosceles triangle with its equal length sides divided into n equal parts with all diagonals drawn, as in A332953. Then T(n,k) = number of k-sided polygons in that figure for k>=3.

Original entry on oeis.org

1, 5, 14, 3, 1, 29, 19, 4, 50, 66, 9, 81, 164, 12, 134, 313, 37, 2, 219, 546, 60, 7, 359, 853, 112, 9, 556, 1294, 160, 16, 1, 779, 1940, 283, 43, 3, 1105, 2780, 360, 53, 6, 1540, 3750, 670, 91, 5, 1, 2087, 5064, 873, 132, 11, 2806, 6625, 1144, 164, 7, 3

Offset: 1

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

See the links in A332953 for images of the triangles.

			Table begins:
1;
5;
14, 3, 1;
29, 19, 4;
50, 66, 9;
81, 164, 12;
134, 313, 37, 2;
219, 546, 60, 7;
359, 853, 112, 9;
556, 1294, 160, 16, 1;
779, 1940, 283, 43, 3;
1105, 2780, 360, 53, 6;
1540, 3750, 670, 91, 5, 1;
2087, 5064, 873, 132, 11;
2806, 6625, 1144, 164, 7, 3;
The row sums are A332953.

Lars Blomberg, Table of n, a(n) for n = 1..613 (the first 70 rows)

Cf. A332953 (regions), A333026 (vertices), A333027 (edges), A007678, A092867, A331452, A331911, A332357, A332358.

A333027 The number of edges formed on an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

Original entry on oeis.org

3, 10, 33, 96, 235, 486, 933, 1600, 2561, 3884, 5907, 8310, 11793, 15890, 20863, 27002, 35229, 44117, 55820, 68312, 82931, 100368, 121711, 143685, 169750, 199509, 232366, 268169, 312132, 355839, 409902, 465503, 527080, 596443, 668961, 746443, 839830, 937967

Offset: 1

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

nonn

See the links in A332953 for images of the triangles.

Lars Blomberg, Table of n, a(n) for n = 1..70

Cf. A332953 (regions), A333025 (n-gons), A333026 (vertices), A007678, A092867, A331452, A331911, A332357, A332358.

a(16) and beyond from Lars Blomberg, May 26 2020

A333274 Irregular triangle read by rows: consider the graph defined in A306302 formed from a row of n adjacent congruent rectangles by drawing the diagonals of all visible rectangles; T(n,k) (n >= 1, 2 <= k <= 2n+2) is the number of vertices in the graph at which k polygons meet.

Original entry on oeis.org

4, 0, 1, 0, 4, 8, 0, 1, 0, 0, 28, 4, 2, 0, 1, 0, 0, 54, 4, 14, 0, 2, 0, 1, 0, 0, 124, 0, 22, 8, 2, 0, 2, 0, 1, 0, 0, 214, 0, 32, 4, 20, 0, 2, 0, 2, 0, 1, 0, 0, 382, 0, 50, 0, 26, 12, 2, 0, 2, 0, 2, 0, 1, 0, 0, 598, 0, 102, 0, 18, 4, 26, 0, 2, 0, 2, 0, 2, 0, 1

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 14 2020

For vertices not on the boundary, the number of polygons meeting at a vertex is simply the degree (or valency) of that vertex.
Row sums are A331755.
Sum_k k*T(n,k) gives A333276.
See A333275 for the degrees of the non-boundary vertices.
Row n is the sum of [0, 0, ..., 0 (n-1 0's), 4, 2*n-2, 0, 0, ..., 0 (n 0's)] and row n of A333275.

			Led d denote the number of polygons meeting at a vertex (except for boundary points, d is the degree of the vertex).
For n=2, the 4 corners have d=3, and on the center line there are 2 vertices with d=4 and 1 with d=6. In the interiors of each of the two squares there are 3 points with d=4.
So in total there are 4 points with d=3, 8 with d=4, and 1 with d=6. So row 2 of the triangle is [0, 4, 8, 0, 1].
The triangle begins:
4,0,1,
0,4,8,0,1,
0,0,28,4,2,0,1,
0,0,54,4,14,0,2,0,1,
0,0,124,0,22,8,2,0,2,0,1,
0,0,214,0,32,4,20,0,2,0,2,0,1;
0,0,382,0,50,0,26,12,2,0,2,0,2,0,1;
0,0,598,0,102,0,18,4,26,0,2,0,2,0,2,0,1;
0,0,950,0,126,0,32,0,30,16,2,0,2,0,2,0,2,0,1;
0,0,1334,0,198,0,62,0,20,4,32,0,2,0,2,0,2,0,2,0,1;
0,0,1912,0,286,0,100,0,10,0,34,20,2,0,2,0,2,0,2,0,2,0,1;
0,0,2622,0,390,0,118,0,38,0,22,4,38,0,2,0,2,0,2,0,2,0,2,0,1;
0,0,3624,0,510,0,136,0,74,0,10,0,38,24,2,0,2,0,2,0,2,0,2,0,2,0,1;
0,0,4690,0,742,0,154,0,118,0,10,0,24,4,44,0,2,0,2,0,2,0,2,0,2,0,2,0,1;

Lars Blomberg, Table of n, a(n) for n = 1..10200 (the first 100 rows)
Lars Blomberg, Pdf printout of Excel spreadsheet showing first 100 rows
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, Image of the vertices for n = 3.
Scott R. Shannon, Image of the vertices for n = 5.
Scott R. Shannon, Image of the vertices for n = 8.
Scott R. Shannon, Image of the vertices for n = 10.
Scott R. Shannon, Image of the vertices for n = 14.

Cf. A306302, A331755, A331757, A331452, A333275, A333276, A333277.

A333275 Irregular triangle read by rows: consider the graph defined in A306302 formed from a row of n adjacent congruent rectangles by drawing the diagonals of all visible rectangles; T(n,k) (n >= 1, 2 <= k <= 2n+2) is the number of non-boundary vertices in the graph at which k polygons meet.

Original entry on oeis.org

0, 0, 1, 0, 0, 6, 0, 1, 0, 0, 24, 0, 2, 0, 1, 0, 0, 54, 0, 8, 0, 2, 0, 1, 0, 0, 124, 0, 18, 0, 2, 0, 2, 0, 1, 0, 0, 214, 0, 32, 0, 10, 0, 2, 0, 2, 0, 1, 0, 0, 382, 0, 50, 0, 22, 0, 2, 0, 2, 0, 2, 0, 1, 0, 0, 598, 0, 102, 0, 18, 0, 12, 0, 2, 0, 2, 0, 2, 0, 1

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 14 2020

The number of polygons meeting at a non-boundary vertex is simply the degree (or valency) of that vertex.
Row sums are A159065.
Sum_k k*T(n,k) gives A333277.
See A333274 for the degrees if the boundary vertices are included.
T(n,k) = 0 if k is odd. But the triangle includes those zero entries because this is used to construct A333274.

			Led d denote the number of polygons meeting at a vertex.
For n=2, in the interiors of each of the two squares there are 3 points with d=4, and the center point has d=6.
So in total there are 6 points with d=4 and 1 with d=6. So row 2 of the triangle is [0, 0, 6, 0, 1].
The triangle begins:
0,0,1,
0,0,6,0,1,
0,0,24,0,2,0,1,
0,0,54,0,8,0,2,0,1,
0,0,124,0,18,0,2,0,2,0,1,
0,0,214,0,32,0,10,0,2,0,2,0,1,
0,0,382,0,50,0,22,0,2,0,2,0,2,0,1,
0,0,598,0,102,0,18,0,12,0,2,0,2,0,2,0,1
...
If we leave out the uninteresting zeros, the triangle begins:
[1]
[6, 1]
[24, 2, 1]
[54, 8, 2, 1]
[124, 18, 2, 2, 1]
[214, 32, 10, 2, 2, 1]
[382, 50, 22, 2, 2, 2, 1]
[598, 102, 18, 12, 2, 2, 2, 1]
[950, 126, 32, 26, 2, 2, 2, 2, 1]
[1334, 198, 62, 20, 14, 2, 2, 2, 2, 1]
[1912, 286, 100, 10, 30, 2, 2, 2, 2, 2, 1]
[2622, 390, 118, 38, 22, 16, 2, 2, 2, 2, 2, 1]
... - _N. J. A. Sloane_, Jul 27 2020

Lars Blomberg, Table of n, a(n) for n = 1..10200 (the first 100 rows)
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, 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

Cf. A306302, A331755, A331757, A331452, A333274, A333276, A333277, A334694.

a(36) and beyond from Lars Blomberg, Jun 17 2020

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

A333543 Irregular triangle read by rows: T(n,k) (n >= 1, k >= n+1) is the number of cells with k vertices in the dissection of an n-dimensional cube by all the hyperplanes that pass through any n vertices.

Original entry on oeis.org

1, 4, 72, 24, 162816, 96576, 118464, 64896, 45888, 22464, 19776, 11904, 8640, 8448, 6144, 1728, 1152, 384, 384, 384

Offset: 1

N. J. A. Sloane, Apr 21 2020

Rows 1 through 4 computed by Veit Elser, later confirmed by Tom Karzes.

The row sums give A333539.

			The two diagonals of a square cut it into four triangular pieces, so T(2,3) = 4.
Triangle begins:
1,
4,
72, 24,
162816, 96576, 118464, 64896, 45888, 22464, 19776, 11904, 8640, 8448, 6144, 1728, 1152, 384, 384, 384,
...

N. J. A. Sloane, Cutting Up a Cube, Math Fun Mailing List, Apr 10 2020; with replies from Tom Karzes, Tomas Rokicki, Veit Elser, and others.

Veit Elser, Rows 1 through 4
Scott R. Shannon, Illustration for a(2) = 4.
Scott R. Shannon, Illustration for a(3) = 72. This shows the 4-faced cells in the 3D cube dissection. The 72 pieces have been moved away from the origin a distance proportional to the average distance of their vertices from the origin.
Scott R. Shannon, Illustration for a(4) = 24. This shows the 5-faced cells in the 3D cube dissection. The 24 pieces have been moved away from the origin a distance proportional to the average distance of their vertices from the origin. These polyhedra form a perfect octahedron inside the original cube with its points touching the cube's inner surface.

Cf. A333539, A333540, A333544, A338622 (number of k-faced polyhedra for the 3D Platonic solids).

For the number of hyperplanes see A007847.

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

A355801 Irregular table read by rows: T(n,k) is the number of k-sided polygons, for k>=3, in a square when straight line segments connect the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.

Original entry on oeis.org

0, 1, 0, 4, 12, 12, 56, 32, 16, 156, 124, 24, 8, 0, 4, 384, 228, 72, 28, 716, 648, 144, 68, 8, 4, 1312, 1144, 240, 112, 8, 2244, 1912, 528, 256, 3528, 3072, 696, 360, 16, 5012, 5536, 1296, 524, 48, 28, 7696, 6596, 1960, 572, 16, 10340, 11448, 2968, 1028, 160, 24, 14520, 14428, 3872, 1156, 104, 8

Offset: 1

Scott R. Shannon, Jul 17 2022

Up to n = 50 the maximum sided k-gon created is the 8-gon. It is plausible this is the maximum sided k-gon for all n, although this is unknown.
See A355798 for more images of the square.
The keyword "look" is for the n = 10 image. - N. J. A. Sloane, Jul 21 2022

			The table begins:
0,     1;
0,     4;
12,    12;
56,    32,    16;
156,   124,   24,   8,    0,   4;
384,   228,   72,   28;
716,   648,   144,  68,   8,   4;
1312,  1144,  240,  112,  8;
2244,  1912,  528,  256;
3528,  3072,  696,  360,  16;
5012,  5536,  1296, 524,  48,  28;
7696,  6596,  1960, 572,  16;
10340, 11448, 2968, 1028, 160, 24;
14520, 14428, 3872, 1156, 104, 8;
19588, 19156, 5296, 2052, 160, 8;
25392, 26112, 7160, 2152, 208, 24;
31820, 37244, 9936, 3240, 488, 64;
.
.

Scott R. Shannon, Image for n = 10.

Cf. A355798 (regions), A355799 (vertices), A355800 (edges), A355801 (k-gons), A255011 (all vertices), A290131, A331452, A335678.

cp's OEIS Frontend

A332953 The number of regions formed inside an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

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A333025 Irregular table read by rows: Take an isosceles triangle with its equal length sides divided into n equal parts with all diagonals drawn, as in A332953. Then T(n,k) = number of k-sided polygons in that figure for k>=3.

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A333027 The number of edges formed on an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

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A333274 Irregular triangle read by rows: consider the graph defined in A306302 formed from a row of n adjacent congruent rectangles by drawing the diagonals of all visible rectangles; T(n,k) (n >= 1, 2 <= k <= 2n+2) is the number of vertices in the graph at which k polygons meet.

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A333275 Irregular triangle read by rows: consider the graph defined in A306302 formed from a row of n adjacent congruent rectangles by drawing the diagonals of all visible rectangles; T(n,k) (n >= 1, 2 <= k <= 2n+2) is the number of non-boundary vertices in the graph at which k polygons meet.

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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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A333543 Irregular triangle read by rows: T(n,k) (n >= 1, k >= n+1) is the number of cells with k vertices in the dissection of an n-dimensional cube by all the hyperplanes that pass through any n vertices.

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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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A355801 Irregular table read by rows: T(n,k) is the number of k-sided polygons, for k>=3, in a square when straight line segments connect the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.

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