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
Examples
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, ...
Links
- 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.
Programs
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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
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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 *)
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
Extensions
T(3,3) corrected and sequence extended by R. J. Mathar, Dec 17 2017
Comments