A331765 -id:A331765 - cp's OEIS frontend

A332600 Triangle read by rows: T(n,k) = number of edges in a "frame" of size n X k (see Comments in A331457 for definition).

8, 28, 92, 80, 240, 360, 178, 508, 604, 860, 372, 944, 1040, 1320, 1792, 654, 1548, 1652, 1956, 2452, 3124, 1124, 2520, 2640, 2968, 3488, 4184, 5256, 1782, 3754, 4004, 4356, 4900, 5620, 6716, 8188, 2724, 5392, 5936, 6312, 6880, 7624, 8744, 10240, 12304, 3914, 7528, 8364, 8764, 9356, 10124, 11268, 12788, 14876, 17460

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 03 2020

See A331457 and A331776 for further illustrations.

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:
[8],
[28, 92],
[80, 240, 360],
[178, 508, 604, 860],
[372, 944, 1040, 1320, 1792],
[654, 1548, 1652, 1956, 2452, 3124],
[1124, 2520, 2640, 2968, 3488, 4184, 5256],
[1782, 3754, 4004, 4356, 4900, 5620, 6716, 8188],
[2724, 5392, 5936, 6312, 6880, 7624, 8744, 10240, 12304],
[3914, 7528, 8364, 8764, 9356, 10124, 11268, 12788, 14876, 17460],
...

Scott R. Shannon, Colored illustration for T(3,3) = 360
Scott R. Shannon, Colored illustration for T(4,4) = 860
N. J. A. Sloane, Illustration for T(3,3) = 360.

Cf. A331457, A331757, A331765, A331776, A332599, A332610, A332611.

The main diagonal is A332597.

Column 1 is A331757, for which there is an explicit formula.
Column 2 is A331765, for which no formula is known.
For m >= n >= 3, T(m,n) = (3*A332610(m,n)+4*A332611(m,n)+4*m+4*n-8)/2, and both A332610 and A332611 have explicit formulas.

More terms from N. J. A. Sloane, Mar 13 2020

A331763 Number of vertices 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).

Original entry on oeis.org

13, 37, 99, 213, 401, 657, 1085, 1619, 2327, 3257, 4457, 5883, 7751, 9885, 12403, 15513, 19131, 23181, 28115, 33601, 39745, 46821, 54865, 63733, 73879, 84889, 97063, 110639, 125649, 141797, 160129, 179981, 201175, 224481, 249403, 276291, 306003, 337425

Offset: 1

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

nonn

Triangles A331452, A331453, A331454 do not have formulas, except for column 1. The column 2 sequences, A331763, A331765, A331766, are therefore the next ones to attack.

See A331452 for other illustrations.

Lars Blomberg, Table of n, a(n) for n = 1..100
Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
Scott R. Shannon, Colored illustration for a(3) = 99
Scott R. Shannon, Data specifically for nX2 (or 2Xn) rectangles
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 20.

Column 2 of A331453.

Cf. A331765, A331766, A331767.

More terms from Scott R. Shannon, Mar 11 2020

a(21) and beyond from Lars Blomberg, Apr 28 2020

A331766 Number of regions 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).

Original entry on oeis.org

16, 56, 142, 296, 544, 892, 1436, 2136, 3066, 4272, 5840, 7688, 10094, 12884, 16182, 20192, 24918, 30200, 36614, 43692, 51756, 61008, 71544, 83040, 96202, 110692, 126702, 144372, 164144, 185200, 209192, 234928, 262706, 293244, 326002, 361240, 400170, 441516

Offset: 1

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

nonn

The grid consists of a rectangular array of 3 X (n+1) dots. If we instead count squares, the dimensions are 2 X n.
Triangles A331452, A331453, A331454 do not have formulas, except for column 1. The column 2 sequences, A331763, A331765, A331766, are therefore the next ones to attack.
See A331452 for other illustrations.
For n<=100, 7-gons: 4 for n=9, 4 for n=18; 8-gons: 2 for n=9; no 9-gons or 10-gons. Lars Blomberg, Apr 28 2020

Lars Blomberg, Table of n, a(n) for n = 1..100
Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
Scott R. Shannon, Colored illustration for a(3) = 142.
Scott R. Shannon, Data specifically for nX2 (or 2Xn) rectangles
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 20.

Column 2 of A331452.

Cf. A331763, A331765.

More terms from Scott R. Shannon, Mar 11 2020

a(21) and beyond from Lars Blomberg, Apr 28 2020

A331454 Triangle read by rows: T(n,m) (n >= m >= 1) = number of line segments formed by drawing the lines connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.

Original entry on oeis.org

8, 28, 92, 80, 240, 596, 178, 508, 1028, 1936, 372, 944, 2004, 3404, 6020, 654, 1548, 3018, 4962, 8064, 11088, 1124, 2520, 4808, 7734, 12708, 17022, 26260, 1782, 3754, 6704, 10840, 16608, 22220, 32794, 42144, 2724, 5392, 9780, 14620, 22788, 30238, 44028, 54024, 72296, 3914, 7528, 12720, 19428, 29914, 37848, 54612, 67590, 86906, 107832

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 27 2020

Take a grid of m+1 X n+1 points. There are 2*(m+n) points on the perimeter. Join every pair of the perimeter points by a line (of finite length). The lines do not extend outside the grid. T(m,n) is the number of line segments formed when these lines intersect each other, and A331452(m,n) and A331453(m,n) give the number of regions and the number of vertices respectively.

For illustrations see the links in A331452.

			Triangle begins:
8,
28, 92,
80, 240, 596,
178, 508, 1028, 1936,
372, 944, 2004, 3404, 6020,
654, 1548, 3018, 4962, 8064, 11088,
1124, 2520, 4808, 7734, 12708, 17022, 26260,
1782, 3754, 6704, 10840, 16608, 22220, 32794, 42144,
2724, 5392, 9780, 14620, 22788, 30238, 44028, 54024, 72296,
...

Lars Blomberg, Table of n, a(n) for n = 1..703 (the first 37 rows)
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.

The main diagonal is A331448.
The first two columns are A331757 and A331765.
Cf. A288187, A331452, A331453.

A330845 The number of edges inside a hexagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

42, 708, 4020, 15120, 38544, 83136, 169686, 294678, 475500, 746340, 1140624, 1581612, 2296986, 3055734, 3980526, 5391264, 7003662, 8516346, 11094810, 13280970, 16180932, 19971282, 24277212, 28090218, 33683862, 39656604, 45901494, 53121744, 62678268, 69382632

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 07 2020

nonn

See the links in A331931 for images of the hexagons.

Cf. A331931 (regions), A331932 (n-gons), A330846 (vertices), A274586 , A332600, A331765.

a(9)-a(30) from Lars Blomberg, May 12 2020

A329710 The number of edges inside a pentagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

20, 290, 2215, 7405, 21150, 43490, 88230, 151135, 250825, 384360, 578840, 814180, 1151525, 1550530, 2063225, 2676925, 3452460, 4333340, 5436210, 6668320, 8154980, 9837690, 11822175, 13993360, 16569650, 19401865, 22636495, 26182350, 30253225, 34608450, 39628050

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 07 2020

nonn

See the links in A331929 for images of the pentagons.

Lars Blomberg, Table of n, a(n) for n = 1..35

Cf. A331929 (regions), A331939 (n-gons), A330847 (vertices), A330845, A274586, A332600, A331765.

a(9) and beyond from Lars Blomberg, May 11 2020

A332419 The number of edges on a decagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

390, 7800, 48870, 164470, 430840, 900890, 1735800, 2982660, 4849740, 7438490, 11017860, 15596420, 21713060, 29254830, 38714410, 50238450, 64311090, 80839300, 100786890, 123786030, 150835530

Offset: 1

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

See the links in A333139 for images of the decagons.

Cf. A333139 (regions), A332417 (n-gons), A332418 (vertices), A330845, A274586, A332600, A331765.

a(6)-a(21) from Lars Blomberg, May 18 2020

A332606 Number of triangles 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).

Original entry on oeis.org

14, 48, 102, 192, 326, 524, 802, 1192, 1634, 2296, 3074, 4052, 5246, 6740, 8398, 10440, 12770, 15512, 18782, 22384, 26386, 31204, 36482, 42232, 48826, 56508, 64318, 73356, 83366, 93996, 106010, 118788, 132634, 148600, 164814, 182648, 201998, 223172, 245634

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.

Lars Blomberg, Table of n, a(n) for n = 1..100
Scott R. Shannon, Data specifically for nX2 (or 2Xn) rectangles

Cf. A331452, A331453, A331454, A331763, A331765, A331766, A332599, A332600, A331457, A332607, A332608, A332609.

a(21) and beyond from Lars Blomberg, Apr 28 2020

A332607 Number of quadrilaterals 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).

Original entry on oeis.org

2, 8, 36, 92, 194, 336, 554, 812, 1314, 1756, 2508, 3252, 4348, 5464, 7054, 8760, 11050, 13324, 16162, 19256, 23188, 27120, 32098, 37396, 43456, 49516, 57608, 65440, 74670, 84388, 95674, 107656, 120990, 133996, 150144, 166424, 185090, 203960, 224926, 247120

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.

Lars Blomberg, Table of n, a(n) for n = 1..100
Scott R. Shannon, Data specifically for nX2 (or 2Xn) rectangles

Cf. A331452, A331453, A331454, A331763, A331765, A331766, A332599, A332600, A331457, A332606, A332608, A332609.

a(21) and beyond from Lars Blomberg, Apr 28 2020

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

cp's OEIS Frontend

A332600 Triangle read by rows: T(n,k) = number of edges in a "frame" of size n X k (see Comments in A331457 for definition).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A331763 Number of vertices 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).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A331766 Number of regions 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).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A331454 Triangle read by rows: T(n,m) (n >= m >= 1) = number of line segments formed by drawing the lines connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A330845 The number of edges inside a hexagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A329710 The number of edges inside a pentagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A332419 The number of edges on a decagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A332606 Number of triangles 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).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A332607 Number of quadrilaterals 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).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A333279 Column 2 of triangle in A288187.

Views

Author

Keywords

Comments

Links

Crossrefs