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 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. 5
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS We may assume n <= m since T(m,n)=T(n,m). LINKS M. A. Alekseyev, M. Basova, 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 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

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Last modified June 15 21:26 EDT 2021. Contains 345051 sequences. (Running on oeis4.)