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A141255 Total number of line segments between points visible to each other in a square n X n lattice. 21
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A line segment joins points (a,b) and (c,d) if the points are distinct and gcd(c-a,d-b)=1.
REFERENCES
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).
LINKS
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
FORMULA
a(n) = A114043(n) - 1.
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
EXAMPLE
The 2 x 2 square lattice has a total of 6 line segments: 2 vertical, 2 horizontal and 2 diagonal.
MATHEMATICA
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}]
(* Seppo Mustonen, May 13 2010 *)
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 *)
PROG
(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
CROSSREFS
Cf. A141224.
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
Sequence in context: A222198 A302650 A055711 * A091321 A125310 A336535
KEYWORD
nonn
AUTHOR
T. D. Noe, Jun 17 2008
STATUS
approved

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Last modified April 21 19:32 EDT 2024. Contains 371885 sequences. (Running on oeis4.)