A333274 -id:A333274

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A331452

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

+0
99

4, 16, 56, 46, 142, 340, 104, 296, 608, 1120, 214, 544, 1124, 1916, 3264, 380, 892, 1714, 2820, 4510, 6264, 648, 1436, 2678, 4304, 6888, 9360, 13968, 1028, 2136, 3764, 6024, 9132, 12308, 17758, 22904, 1562, 3066, 5412, 8126, 12396, 16592, 23604, 29374, 38748, 2256, 4272, 7118, 10792, 16226, 20896, 29488, 36812, 47050, 58256

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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 segment. The lines do not extend outside the grid. T(m,n) is the number of regions formed by these lines, and A331453(m,n) and A331454(m,n) give the number of vertices and the number of line segments respectively.

A288187 is a similar sequence, except there every pair of the (m+1)*(n+1) points of the grid (including the interior points) are joined by line segments. The (m,1) (m>=1) and (2,2) entries here and in A288187 are the same, while all other entries are different.

REFERENCES

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, Integers, Ron Graham Memorial Volume 21A (2021), #A5. Also in book, "Number Theory and Combinatorics: A Collection in Honor of the Mathematics of Ronald Graham", ed. B. M. Landman et al., De Gruyter, 2022, pp. 65-97.

LINKS

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.

Johnny Fonseca, Intersections and Segments, Illustrations for T(n,m) with 2 <= n <= m <= 10, with intersection points shown on the left, and the full structures on the right. Solution to homework problem, Math 640, Rutgers Univ., Feb 11 2020.

Scott R. Shannon, Colored illustration for T(1,1)

Scott R. Shannon, Colored illustration for T(2,1)

Scott R. Shannon, Colored illustration for T(3,1)

Scott R. Shannon, Colored illustration for T(4,1)

Scott R. Shannon, Colored illustration for T(5,1)

Scott R. Shannon, Colored illustration for T(6,1)

Scott R. Shannon, Colored illustration for T(7,1)

Scott R. Shannon, Colored illustration for T(8,1)

Scott R. Shannon, Colored illustration for T(9,1)

Scott R. Shannon, Colored illustration for T(10,1)

Scott R. Shannon, Colored illustration for T(11,1)

Scott R. Shannon, Colored illustration for T(12,1)

Scott R. Shannon, Colored illustration for T(13,1)

Scott R. Shannon, Colored illustration for T(14,1)

Scott R. Shannon, Colored illustration for T(15,1)

Scott R. Shannon, Colored illustration for T(2,2)

Scott R. Shannon, Colored illustration for T(3,2)

Scott R. Shannon, Colored illustration for T(4,2)

Scott R. Shannon, Colored illustration for T(5,2)

Scott R. Shannon, Colored illustration for T(6,2)

Scott R. Shannon, Colored illustration for T(9,2)

Scott R. Shannon, Colored illustration for T(9,2) (edge number coloring)

Scott R. Shannon, Colored illustration for T(10,2)

Scott R. Shannon, Colored illustration for T(10,2) (edge number coloring)

Scott R. Shannon, Colored illustration for T(3,3)

Scott R. Shannon, Colored illustration for T(4,3)

Scott R. Shannon, Colored illustration for T(5,3)

Scott R. Shannon, Colored illustration for T(6,3)

Scott R. Shannon, Colored illustration for T(9,3)

Scott R. Shannon, Colored illustration for T(11,3) [The top of the figure has been modified]

Scott R. Shannon, Colored illustration for T(4,4)

Scott R. Shannon, Colored illustration for T(5,4)

Scott R. Shannon, Colored illustration for T(6,4)

Scott R. Shannon, Colored illustration for T(5,5)

Scott R. Shannon, Colored illustration for T(6,5)

Scott R. Shannon, Colored illustration for T(6,6)

Scott R. Shannon, Colored illustration for T(6,6) (another version)

Scott R. Shannon, Colored illustration for T(7,7)

Scott R. Shannon, Colored illustration for T(10,7)

Scott R. Shannon, Data underlying this triangle and A331453, A331454 [Includes numbers of polygonal regions with each number of edges.]

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, 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)

EXAMPLE

Triangle begins:

16, 56;

46, 142, 340;

104, 296, 608, 1120;

214, 544, 1124, 1916, 3264;

380, 892, 1714, 2820, 4510, 6264;

648, 1436, 2678, 4304, 6888, 9360, 13968;

1028, 2136, 3764, 6024, 9132, 12308, 17758, 22904;

1562, 3066, 5412, 8126, 12396, 16592, 23604, 29374, 38748;

2256, 4272, 7118, 10792, 16226, 20896, 29488, 36812, 47050, 58256;

...

CROSSREFS

The first column is A306302, the main diagonal is A255011.

The second column is A331766.

See A333274 for the classification of vertices by valency.

Cf. A288187, A331453, A331454, A333286, A333287, A333288.

KEYWORD

nonn,tabl,nice

AUTHOR

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

STATUS

approved

A331755

Number of vertices in a regular drawing of the complete bipartite graph K_{n,n}.

+0
28

2, 5, 13, 35, 75, 159, 275, 477, 755, 1163, 1659, 2373, 3243, 4429, 5799, 7489, 9467, 11981, 14791, 18275, 22215, 26815, 31847, 37861, 44499, 52213, 60543, 70011, 80347, 92263, 105003, 119557, 135327, 152773, 171275, 191721, 213547, 237953

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..1000

Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.

M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5, Lemma 2.

S. Legendre, The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph, J. Integer Seqs., Vol. 12, 2009.

Scott R. Shannon, Images of vertices for n=2.

Scott R. Shannon, Images of vertices for n=3.

Scott R. Shannon, Images of vertices for n=4.

Scott R. Shannon, Images of vertices for n=5.

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=10.

Scott R. Shannon, Images of vertices for n=12.

Scott R. Shannon, Images of vertices for n=15.

Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Index entries for sequences related to stained glass windows

FORMULA

a(n) = A290132(n) - A290131(n) + 1.

a(n) = A159065(n) + 2*n.

This is column 1 of A331453.

a(n) = (9/(8*Pi^2))*n^4 + O(n^3 log(n)). Asymptotic to (9/(2*Pi^2))*A000537(n-1). [Stéphane Legendre, see A159065.]

MAPLE

# Maple code from N. J. A. Sloane, Jul 16 2020

V106i := proc(n) local ans, a, b; ans:=0;

for a from 1 to n-1 do for b from 1 to n-1 do

if igcd(a, b)=1 then ans:=ans + (n-a)*(n-b); fi; od: od: ans; end; # A115004

V106ii := proc(n) local ans, a, b; ans:=0;

for a from 1 to floor(n/2) do for b from 1 to floor(n/2) do

if igcd(a, b)=1 then ans:=ans + (n-2*a)*(n-2*b); fi; od: od: ans; end; # A331761

A331755 := n -> 2*(n+1) + V106i(n+1) - V106ii(n+1);

MATHEMATICA

a[n_]:=Module[{x, y, s1=0, s2=0}, For[x=1, x<=n-1, x++, For[y=1, y<=n-1, y++, If[GCD[x, y]==1, s1+=(n-x)*(n-y); If[2*x<=n-1&&2*y<=n-1, s2+=(n-2*x)*(n-2*y)]]]]; s1-s2]; Table[a[n]+ 2 n, {n, 1, 40}] (* Vincenzo Librandi, Feb 04 2020 *)

CROSSREFS

Cf. A290131 (regions), A290132 (edges), A333274 (polygons per vertex), A333276, A159065.

For K_n see A007569, A007678, A135563.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 02 2020

STATUS

approved

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.

+0
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

COMMENTS

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.

LINKS

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

EXAMPLE

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

CROSSREFS

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

KEYWORD

nonn,tabf

AUTHOR

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

EXTENSIONS

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

STATUS

approved

A333276

a(n) = Sum_k k*A333274(n,k).

+0
5

12, 50, 152, 346, 732, 1294, 2232, 3546, 5428, 7806, 11136, 15226, 20676, 27150, 35048, 44386, 56044, 69302, 85480, 103882, 125180, 148942, 176968, 208034, 243772, 283014, 327272, 375826, 431212, 490918, 558456, 631978, 712844, 799726, 895152, 997322, 1110628

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(n)/A331755(n) is the average number of polygons touching a vertex in the graph defined in A306302.

LINKS

Lars Blomberg, Table of n, a(n) for n = 1..100

Scott R. Shannon, Images of vertices for n=1.

Scott R. Shannon, Images of vertices for n=2.

Scott R. Shannon, Images of vertices for n=3.

Scott R. Shannon, Images of vertices for n=4.

Scott R. Shannon, Images of vertices for n=9.

Scott R. Shannon, Images of vertices for n=14.

CROSSREFS

Cf. A159065, A306302, A333274, A333275, A333277.

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

a(15) and beyond from Lars Blomberg, Jun 17 2020

STATUS

approved

A333277

a(n) = Sum_k k*A333275(n,k).

+0
4

4, 30, 116, 290, 652, 1186, 2092, 3370, 5212, 7546, 10828, 14866, 20260, 26674, 34508, 43778, 55364, 68546, 84644, 102962, 124172, 147842, 175772, 206738, 242372, 281506, 325652, 374090, 429356, 488938, 556348, 629738, 710468, 797210, 892492, 994514, 1107668

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(n)/A331755(n) is the average number of polygons touching a non-boundary vertex in the graph defined in A306302.

LINKS

Lars Blomberg, Table of n, a(n) for n = 1..100

CROSSREFS

Cf. A159065, A306302, A333274, A333275, A333276.

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

a(6) and beyond from Lars Blomberg, Jun 17 2020

STATUS

approved

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.

+0
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

It would be nice to have a formula or recurrence. - N. J. A. Sloane, Jun 22 2020

LINKS

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