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%I #34 Dec 05 2022 20:49:51
%S 0,6,28,86,200,418,748,1282,2040,3106,4492,6394,8744,11822,15556,
%T 20074,25456,32086,39724,48934,59456,71554,85252,101250,119040,139350,
%U 161932,187254,215136,246690,280916,319346,361328,407302,457180,511714,570232
%N Total number of line segments between points visible to each other in a square n X n lattice.
%C A line segment joins points (a,b) and (c,d) if the points are distinct and gcd(c-a,d-b)=1.
%D D. M. Acketa, J. D. Zunic: On the number of linear partitions of the (m,n)-grid. Inform. Process. Lett., 38 (3) (1991), 163-168. See Table A.1.
%D Jovisa Zunic, Note on the number of two-dimensional threshold functions, SIAM J. Discrete Math. Vol. 25 (2011), No. 3, pp. 1266-1268. See Eq. (1.2).
%H Chai Wah Wu, <a href="/A141255/b141255.txt">Table of n, a(n) for n = 1..10000</a>
%H Seppo Mustonen, <a href="http://www.survo.fi/papers/LinesInGrid2.pdf">On lines going through a given number of points in a rectangular grid of points</a> [From _Seppo Mustonen_, May 13 2010]
%H Seppo Mustonen, <a href="/A141255/a141255.pdf">On lines going through a given number of points in a rectangular grid of points</a> [Local copy]
%H N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)
%F a(n) = A114043(n) - 1.
%F a(n) = 2*(n-1)*(2n-1) + 2*Sum_{i=2..n-1} (n-i)*(2n-i)*phi(i). - _Chai Wah Wu_, Aug 16 2021
%e The 2 x 2 square lattice has a total of 6 line segments: 2 vertical, 2 horizontal and 2 diagonal.
%t Table[cnt=0; Do[If[GCD[c-a,d-b]<2, cnt++ ], {a,n}, {b,n}, {c,n}, {d,n}]; (cnt-n^2)/2, {n,20}]
%t (* This recursive code is much more efficient. *)
%t a[n_]:=a[n]=If[n<=1,0,2*a1[n]-a[n-1]+R1[n]]
%t a1[n_]:=a1[n]=If[n<=1,0,2*a[n-1]-a1[n-1]+R2[n]]
%t R1[n_]:=R1[n]=If[n<=1,0,R1[n-1]+4*EulerPhi[n-1]]
%t R2[n_]:=(n-1)*EulerPhi[n-1]
%t Table[a[n],{n,1,37}]
%t (* _Seppo Mustonen_, May 13 2010 *)
%t a[n_]:=2 Sum[(n-i) (n-j) Boole[CoprimeQ[i,j]], {i,1,n-1}, {j,1,n-1}] + 2 n^2 - 2 n; Array[a, 40] (* _Vincenzo Librandi_, Feb 05 2020 *)
%o (Python)
%o from sympy import totient
%o def A141255(n): return 2*(n-1)*(2*n-1) + 2*sum(totient(i)*(n-i)*(2*n-i) for i in range(2,n)) # _Chai Wah Wu_, Aug 16 2021
%Y Cf. A141224.
%Y 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
%K nonn
%O 1,2
%A _T. D. Noe_, Jun 17 2008
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