A107348 Triangle read by rows: T(m,n) = number of different lines in a rectangular m X n array of points with integer coordinates (x,y): 0 <= x <= m, 0 <= y <= n.

0, 1, 6, 1, 11, 20, 1, 18, 35, 62, 1, 27, 52, 93, 140, 1, 38, 75, 136, 207, 306, 1, 51, 100, 181, 274, 405, 536, 1, 66, 131, 238, 361, 534, 709, 938, 1, 83, 164, 299, 454, 673, 894, 1183, 1492, 1, 102, 203, 370, 563, 836, 1111, 1470, 1855, 2306

Offset

0,3

Comments

We may assume n <= m since T(m,n)=T(n,m).

Links

Table of n, a(n) for n=0..54.

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh, On the minimal teaching sets of two-dimensional threshold functions, SIAM J. Disc. Math. 29(1), 2015, pp. 157-165.

Les Reid, Problem #7: How Many Lines Does the Lattice of Points Generate?, Problems from the 04-05 academic year, Challenge Archive, Missouri State University's Problem Corner.

Formula

T(0, 0) = 0; T(m, 0) = 1, m >= 1.

When both m,n -> +oo, T(m,n) / 2Cmn -> 9/(2*pi^2). - Dan Dima, Mar 18 2006

T(n,m) = A295707(n,m). - R. J. Mathar, Dec 17 2017

Example

Triangle begins
0,
1, 6,
1, 11, 20,
1, 18, 35, 62,
1, 27, 52, 93, 140,
1, 38, 75, 136, 207, 306,
1, 51, 100, 181, 274, 405, 536,
1, 66, 131, 238, 361, 534, 709, 938,
1, 83, 164, 299, 454, 673, 894, 1183, 1492,
1, 102, 203, 370, 563, 836, 1111, 1470, 1855, 2306,
...

Maple

VR := proc(m, n, q) local a, i, j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i, j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
LL:=(m, n)->(VR(m, n, 1)-VR(m, n, 2))/2;
for m from 1 to 12 do lprint([seq(LL(m, n), n=1..m)]); od: # N. J. A. Sloane, Feb 10 2020

Mathematica

VR[m_, n_, q_] := Sum[If[GCD[i, j] == q, (m - Abs[i])(n - Abs[j]), 0], {i, -m + 1, m - 1}, {j, -n + 1, n - 1}];
LL[m_, n_] := (1/2)(VR[m, n, 1] - VR[m, n, 2]);
Table[LL[m, n], {m, 1, 10}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jun 04 2023, after N. J. A. Sloane *)

Crossrefs

Cf. A295707 (symmetric array), A018808 (diagonal). A160842 - A160850 (columns).

Sequence in context: A328898 A216605 A342635 * A342917 A040041 A162933

Adjacent sequences: A107345 A107346 A107347 * A107349 A107350 A107351

Keyword

nonn,tabl

Author

Dan Dima, May 23 2005

Extensions

T(3,3) corrected and sequence extended by R. J. Mathar, Dec 17 2017

Status

approved