A290132 -id:A290132 - cp's OEIS frontend

A290131 Number of regions in a regular drawing of the complete bipartite graph K_{n,n}.

0, 2, 12, 40, 96, 204, 368, 634, 1012, 1544, 2236, 3186, 4360, 5898, 7764, 10022, 12712, 16026, 19844, 24448, 29708, 35756, 42604, 50602, 59496, 69650, 80940, 93600, 107540, 123316, 140428, 159642, 180632, 203618, 228556, 255822, 285080, 317326, 352020, 389498

Offset: 1

R. J. Mathar, Jul 20 2017

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Martin Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5. See Lemma 2 and Table 1.
Stéphane Legendre, The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph, J. Integer Seqs., Vol. 12, 2009.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Cf. A115004, A159065, A290132, A331754.
For K_n see A007569, A007678, A135563.
The following eight sequences are all essentially the same. The simplest is A115004(n), which we denote by z(n). Then A088658(n) = 4*z(n-1); A114043(n) = 2*z(n-1)+2*n^2-2*n+1; A114146(n) = 2*A114043(n); A115005(n) = z(n-1)+n*(n-1); A141255(n) = 2*z(n-1)+2*n*(n-1); A290131(n) = z(n-1)+(n-1)^2; A306302(n) = z(n)+n^2+2*n. - N. J. A. Sloane, Feb 04 2020

A290131 := proc(n)
    A115004(n-1)+(n-1)^2 ;
end proc:
seq(A290131(n),n=1..30) ;

z[n_] := Sum[(n - i + 1)(n - j + 1) Boole[GCD[i, j] == 1], {i, n}, {j, n}];
a[n_] := z[n - 1] + (n - 1)^2;
Array[a, 40] (* Jean-François Alcover, Mar 24 2020 *)

from math import gcd
def a115004(n):
    r=0
    for a in range(1, n + 1):
        for b in range(1, n + 1):
            if gcd(a, b)==1:r+=(n + 1 - a)*(n + 1 - b)
    return r
def a(n): return a115004(n - 1) + (n - 1)**2
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 20 2017, after Maple code

from sympy import totient
def A290131(n): return 2*(n-1)**2 + sum(totient(i)*(n-i)*(2*n-i) for i in range(2,n)) # Chai Wah Wu, Aug 16 2021

a(n) = A115004(n-1) + (n-1)^2.

a(n) = 2*(n-1)^2 + Sum_{i=2..n-1} (n-i)*(2n-i)*phi(i). - Chai Wah Wu, Aug 16 2021

A331755 Number of vertices in a regular drawing of the complete bipartite graph K_{n,n}.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Feb 02 2020

nonn

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

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

For K_n see A007569, A007678, A135563.

# 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);

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 *)

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.]

A359693 Number of edges in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.

Original entry on oeis.org

6, 24, 162, 670, 4456, 8942, 44470, 98902, 259114, 438552, 1330566, 1897164, 4893752, 7246502, 11544278, 17678880

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 11 2023

The number of vertices along each edge is A005728(n). No formula for a(n) is known.

See A359690 and A359692 for images of the graph.

Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Wikipedia, Farey sequence.

Cf. A359690 (vertices), A359691 (crossings), A359692 (regions), A359694 (k-gons), A005728, A290132, A359655, A358888, A358884, A006842, A006843.

a(n) = A359690(n) + A359692(n) - 2*A005728(n) + 1 by Euler's formula.

A369177 Number of edges in a graph of n adjacent rectangles in a row with all possible diagonals drawn, as in A306302, but without the rectangles' edges which are perpendicular to the row.

Original entry on oeis.org

6, 24, 66, 152, 318, 576, 998, 1608, 2474, 3600, 5162, 7100, 9670, 12772, 16546, 21036, 26622, 33024, 40814, 49716, 60006, 71560, 85158, 100264, 117626, 136780, 158358, 182080, 209106, 238312, 271314, 307304, 346866, 389488, 436286, 486444, 542026, 601436, 665814, 734504, 809882, 889544

Offset: 1

Scott R. Shannon, Jan 15 2024

nonn

See A369175 and A369176 for images of the graph.

Cf. A369175 (regions), A369176 (vertices), A369178 (k-gons), A306302, A331757, A290132, A368757.

a(n) = A369175(n) + A369176(n) - 1 by Euler's formula.

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

A290131 Number of regions in a regular drawing of the complete bipartite graph K_{n,n}.

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A331755 Number of vertices in a regular drawing of the complete bipartite graph K_{n,n}.

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A359693 Number of edges in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.

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A369177 Number of edges in a graph of n adjacent rectangles in a row with all possible diagonals drawn, as in A306302, but without the rectangles' edges which are perpendicular to the row.

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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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