A331763 -id:A331763 - cp's OEIS frontend

A335687 (A331763(n) - A331755(n+1))/2.

4, 12, 32, 69, 121, 191, 304, 432, 582, 799, 1042, 1320, 1661, 2043, 2457, 3023, 3575, 4195, 4920, 5693, 6465, 7487, 8502, 9617, 10833, 12173, 13526, 15146, 16693, 18397, 20286, 22327, 24201, 26603, 28841, 31372, 34025, 36873, 39583, 42913, 46029

Offset: 1

N. J. A. Sloane, Jul 14 2020

nonn

One-half of ((number of vertices in graph SC(n,2)) - (number of vertices in graph SC(n,1))).
It would be nice to have a formula for this sequence. The graphs SC(n,1) are fairly well understood, while SC(n,m) is basically a mystery for m >= 2.
Note that the offsets in A331755 and A331763 have different meanings, which is why there is an extra "+1" in the definition of the current sequence.

			For n=2, SC(2,2) has 37 vertices and SC(2,1) has 13 vertices (see illustrations), so a(2) = (37-13)/2 = 12.

N. J. A. Sloane, Table of n, a(n) for n = 1..100
Scott R. Shannon, Colored illustration for the graph SC(2,1)
Scott R. Shannon, Colored illustration for the graph SC(2,2)

Cf. A331452, A331755, A331763.

A331765 Number of edges 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

28, 92, 240, 508, 944, 1548, 2520, 3754, 5392, 7528, 10296, 13570, 17844, 22768, 28584, 35704, 44048, 53380, 64728, 77292, 91500, 107828, 126408, 146772, 170080, 195580, 223764, 255010, 289792, 326996, 369320, 414908, 463880, 517724, 575404, 637530, 706172

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
Scott R. Shannon, Colored illustration for a(3) = 240.
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.

Column 2 of A331454.

Cf. A331763, A331766.

More terms from Scott R. Shannon, Mar 11 2020

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

A332599 Triangle read by rows: T(n,k) = number of vertices in a "frame" of size n X k (see Comments in A331457 for definition).

Original entry on oeis.org

5, 13, 37, 35, 99, 152, 75, 213, 256, 364, 159, 401, 448, 568, 776, 275, 657, 704, 836, 1056, 1340, 477, 1085, 1132, 1276, 1508, 1804, 2272, 755, 1619, 1712, 1868, 2112, 2420, 2900, 3532, 1163, 2327, 2552, 2720, 2976, 3296, 3788, 4432, 5336, 1659, 3257, 3568, 3748, 4016, 4348, 4852, 5508, 6424, 7516

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:
[5],
[13, 37],
[35, 99, 152],
[75, 213, 256, 364],
[159, 401, 448, 568, 776],
[275, 657, 704, 836, 1056, 1340],
[477, 1085, 1132, 1276, 1508, 1804, 2272],
[755, 1619, 1712, 1868, 2112, 2420, 2900, 3532],
[1163, 2327, 2552, 2720, 2976, 3296, 3788, 4432, 5336],
[1659, 3257, 3568, 3748, 4016, 4348, 4852, 5508, 6424, 7516],
...

Scott R. Shannon, Colored illustration for T(3,3) = 152
Scott R. Shannon, Colored illustration for T(4,4) = 364
N. J. A. Sloane, Illustration for T(3,3) = 152.

Cf. A331457, A331755, A331763, A331776, A332599, A332600.

The main diagonal is A332598.

Column 1 is A331755, for which there is an explicit formula.
Column 2 is A331763, for which no formula is known.
For m >= n >= 3, T(m,n) = A332600(m,n) - A331457(m,n) (Euler for genus 1 graph), and both A332600 and A331457 have explicit formulas.

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

A331453 Triangle read by rows: T(n,m) (n >= m >= 1) = number of vertices 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

5, 13, 37, 35, 99, 257, 75, 213, 421, 817, 159, 401, 881, 1489, 2757, 275, 657, 1305, 2143, 3555, 4825, 477, 1085, 2131, 3431, 5821, 7663, 12293, 755, 1619, 2941, 4817, 7477, 9913, 15037, 19241, 1163, 2327, 4369, 6495, 10393, 13647, 20425, 24651, 33549, 1659, 3257, 5603, 8637, 13689, 16953, 25125, 30779, 39857, 49577

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 vertices in the resulting diagram, and A331452(m,n) and A331454(m,n) give the number of regions and the number of line segments respectively.

For illustrations see the links in A331452.

			Triangle begins:
5,
13, 37,
35, 99, 257,
75, 213, 421, 817,
159, 401, 881, 1489, 2757,
275, 657, 1305, 2143, 3555, 4825,
477, 1085, 2131, 3431, 5821, 7663, 12293,
755, 1619, 2941, 4817, 7477, 9913, 15037, 19241,
1163, 2327, 4369, 6495, 10393, 13647, 20425, 24651, 33549,
...

Lars Blomberg, Table of n, a(n) for n = 1..703 (the first 37 rows)
Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
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, Conant's Gasket, Recamán Variations, the Enots Wolley Sequence, and Stained Glass Windows, Experimental Math Seminar, Rutgers University, Sep 10 2020 (video of Zoom talk)

The main diagonal is A331449.
The first two columns are A331755 and A331763.
Cf. A288187, A331452, A331454.

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

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

A333281 Column 2 of triangle in A288180.

Original entry on oeis.org

13, 37, 121, 265, 587, 1019, 1797, 2823, 4369, 6257, 9001, 12289, 16775, 21905, 28383, 35901, 45463, 56119, 69351, 84167, 101687, 120869, 143777, 168873, 198191, 229771, 266015, 305379, 350673, 399035, 454243, 513619, 579787, 649899, 727927, 810907, 903581

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

A335687 (A331763(n) - A331755(n+1))/2.

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A331765 Number of edges 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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A332599 Triangle read by rows: T(n,k) = number of vertices in a "frame" of size n X k (see Comments in A331457 for definition).

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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).

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A331453 Triangle read by rows: T(n,m) (n >= m >= 1) = number of vertices formed by drawing the lines connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.

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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).

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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).

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A333279 Column 2 of triangle in A288187.

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A333280 Column 2 of triangle in A333278.

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A333281 Column 2 of triangle in A288180.

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