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 A332367 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1*x_1 + a_2*x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = number of triangular cells in the partition for m >= n >= 2. 6
 4, 8, 20, 12, 32, 52, 16, 48, 80, 124, 20, 64, 108, 168, 228, 24, 84, 144, 228, 312, 428, 28, 104, 180, 288, 396, 544, 692, 32, 128, 224, 360, 496, 684, 872, 1100, 36, 152, 268, 432, 596, 824, 1052, 1328, 1604, 40, 180, 320, 520, 720, 1000, 1280, 1620, 1960, 2396 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 LINKS Table of n, a(n) for n=2..56. M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Theorem 12. N. J. A. Sloane, Illustration for m=n=3 EXAMPLE Triangle begins: 4, 8, 20, 12, 32, 52, 16, 48, 80, 124, 20, 64, 108, 168, 228, 24, 84, 144, 228, 312, 428, 28, 104, 180, 288, 396, 544, 692, 32, 128, 224, 360, 496, 684, 872, 1100, 36, 152, 268, 432, 596, 824, 1052, 1328, 1604, ... MAPLE # Maple code for sequences mentioned in Theorem 12 of Alekseyev et al. (2015). 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; VS := proc(m, n) local a, i, j; a:=0; # A331781 for i from 1 to m-1 do for j from 1 to n-1 do if gcd(i, j)=1 then a:=a+1; fi; od: od: a; end; c3 := (m, n) -> VR(m, n, 2)+4; # A332367 for m from 2 to 12 do lprint([seq(c3(m, n), n=2..m)]); od: [seq(c3(n, n)/4, n=2..40)]; # A332368 c4 := (m, n) -> VR(m, n, 1)/2 - VR(m, n, 2) - 3; # A332369 for m from 2 to 12 do lprint([seq(c4(m, n), n=2..m)]); od: [seq(c4(n, n), n=2..40)]; # A332370 ct := (m, n) -> c3(m, n)+c4(m, n); # A332371 for m from 2 to 12 do lprint([seq(ct(m, n), n=2..m)]); od: [seq(ct(n, n), n=2..40)]; # A114043 et := (m, n) -> VR(m, n, 1) - VR(m, n, 2)/2 - VS(m, n) - 2; # A332372 for m from 2 to 12 do lprint([seq(et(m, n), n=2..m)]); od: [seq(et(n, n), n=2..40)]; # A332373 vt := (m, n) -> et(m, n) - ct(m, n) +1; # A332374 for m from 2 to 12 do lprint([seq(vt(m, n), n=2..m)]); od: [seq(vt(n, n), n=2..40)]; # A332375 CROSSREFS Cf. A332350, A332352, A331781, A332371, A332372, A332374. For main diagonal see A332368. Sequence in context: A340948 A265108 A328184 * A273143 A273174 A178447 Adjacent sequences: A332364 A332365 A332366 * A332368 A332369 A332370 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane, Feb 12 2020 STATUS approved

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Last modified May 18 19:36 EDT 2024. Contains 372666 sequences. (Running on oeis4.)