login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A141255
Total number of line segments between points visible to each other in a square n X n lattice.
23
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
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).
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