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A141255
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Total number of line segments between points visible to each other in a square n X n lattice.
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21
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0, 6, 28, 86, 200, 418, 748, 1282, 2040, 3106, 4492, 6394, 8744, 11822, 15556, 20074, 25456, 32086, 39724, 48934, 59456, 71554, 85252, 101250, 119040, 139350, 161932, 187254, 215136, 246690, 280916, 319346, 361328, 407302, 457180, 511714, 570232
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OFFSET
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1,2
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COMMENTS
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A line segment joins points (a,b) and (c,d) if the points are distinct and gcd(c-a,d-b)=1.
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REFERENCES
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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.
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).
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LINKS
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FORMULA
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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
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EXAMPLE
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The 2 x 2 square lattice has a total of 6 line segments: 2 vertical, 2 horizontal and 2 diagonal.
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MATHEMATICA
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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}]
(* This recursive code is much more efficient. *)
a[n_]:=a[n]=If[n<=1, 0, 2*a1[n]-a[n-1]+R1[n]]
a1[n_]:=a1[n]=If[n<=1, 0, 2*a[n-1]-a1[n-1]+R2[n]]
R1[n_]:=R1[n]=If[n<=1, 0, R1[n-1]+4*EulerPhi[n-1]]
R2[n_]:=(n-1)*EulerPhi[n-1]
Table[a[n], {n, 1, 37}]
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 *)
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PROG
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(Python)
from sympy import totient
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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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