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

%I A331755 #67 Feb 16 2025 08:33:59
%S A331755 2,5,13,35,75,159,275,477,755,1163,1659,2373,3243,4429,5799,7489,9467,
%T A331755 11981,14791,18275,22215,26815,31847,37861,44499,52213,60543,70011,
%U A331755 80347,92263,105003,119557,135327,152773,171275,191721,213547,237953
%N A331755 Number of vertices in a regular drawing of the complete bipartite graph K_{n,n}.
%H A331755 N. J. A. Sloane, <a href="/A331755/b331755.txt">Table of n, a(n) for n = 1..1000</a>
%H A331755 Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, <a href="http://neilsloane.com/doc/rose_5.pdf">Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids</a>, (2020). Also arXiv:2009.07918.
%H A331755 M. Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Griffiths2/griffiths.html">Counting the regions in a regular drawing of K_{n,n}</a>, J. Int. Seq. 13 (2010) # 10.8.5, Lemma 2.
%H A331755 S. Legendre, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Legendre/legendre2.html">The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph</a>, J. Integer Seqs., Vol. 12, 2009.
%H A331755 Scott R. Shannon, <a href="/A331755/a331755.png">Images of vertices for n=2</a>.
%H A331755 Scott R. Shannon, <a href="/A331755/a331755_1.png">Images of vertices for n=3</a>.
%H A331755 Scott R. Shannon, <a href="/A331755/a331755_2.png">Images of vertices for n=4</a>.
%H A331755 Scott R. Shannon, <a href="/A331755/a331755_3.png">Images of vertices for n=5</a>.
%H A331755 Scott R. Shannon, <a href="/A331755/a331755_7.png">Images of vertices for n=6</a>
%H A331755 Scott R. Shannon, <a href="/A331755/a331755_8.png">Images of vertices for n=7</a>
%H A331755 Scott R. Shannon, <a href="/A331755/a331755_11.png">Images of vertices for n=8</a>
%H A331755 Scott R. Shannon, <a href="/A331755/a331755_10.png">Images of vertices for n=9</a>
%H A331755 Scott R. Shannon, <a href="/A331755/a331755_4.png">Images of vertices for n=10</a>.
%H A331755 Scott R. Shannon, <a href="/A331755/a331755_5.png">Images of vertices for n=12</a>.
%H A331755 Scott R. Shannon, <a href="/A331755/a331755_6.png">Images of vertices for n=15</a>.
%H A331755 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H A331755 <a href="/index/St#Stained">Index entries for sequences related to stained glass windows</a>
%F A331755 a(n) = A290132(n) - A290131(n) + 1.
%F A331755 a(n) = A159065(n) + 2*n.
%F A331755 This is column 1 of A331453.
%F A331755 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.]
%p A331755 # Maple code from _N. J. A. Sloane_, Jul 16 2020
%p A331755 V106i := proc(n) local ans,a,b; ans:=0;
%p A331755 for a from 1 to n-1 do for b from 1 to n-1 do
%p A331755 if igcd(a,b)=1 then ans:=ans + (n-a)*(n-b); fi; od: od: ans; end; # A115004
%p A331755 V106ii := proc(n) local ans,a,b; ans:=0;
%p A331755 for a from 1 to floor(n/2) do for b from 1 to floor(n/2) do
%p A331755 if igcd(a,b)=1 then ans:=ans + (n-2*a)*(n-2*b); fi; od: od: ans; end; # A331761
%p A331755 A331755 := n -> 2*(n+1) + V106i(n+1) - V106ii(n+1);
%t A331755 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 *)
%Y A331755 Cf. A290131 (regions), A290132 (edges), A333274 (polygons per vertex), A333276, A159065.
%Y A331755 For K_n see A007569, A007678, A135563.
%K A331755 nonn
%O A331755 1,1
%A A331755 _N. J. A. Sloane_, Feb 02 2020