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A141255 Total number of line segments between points visible to each other in a square n X n lattice. 18
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.

LINKS

Table of n, a(n) for n=1..37.

S. Mustonen, On lines going through a given number of points in a rectangular grid of points [From Seppo Mustonen, May 13 2010]

Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points [Local copy]

N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

FORMULA

a(n) = A114043(n) - 1.

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}]

Contribution from Seppo Mustonen, May 13 2010: (Start)

(* 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}]

(End)

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 *)

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

Adjacent sequences:  A141252 A141253 A141254 * A141256 A141257 A141258

KEYWORD

nonn

AUTHOR

T. D. Noe, Jun 17 2008

STATUS

approved

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Last modified June 24 17:51 EDT 2021. Contains 345419 sequences. (Running on oeis4.)