A332361 Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1*x_1 + a_2*x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of vertices in the partition, for m >= n >= 1.
3, 4, 6, 5, 9, 14, 6, 13, 22, 36, 7, 18, 31, 52, 76, 8, 24, 43, 74, 110, 160, 9, 31, 56, 97, 144, 210, 276, 10, 39, 72, 126, 188, 275, 363, 478, 11, 48, 89, 157, 235, 345, 456, 601, 756, 12, 58, 109, 193, 290, 427, 565, 745, 938, 1164, 13, 69, 130, 231, 347, 511, 675, 890, 1120, 1390, 1660
Offset: 1
Examples
Triangle begins: 3, 4, 6, 5, 9, 14, 6, 13, 22, 36, 7, 18, 31, 52, 76, 8, 24, 43, 74, 110, 160, 9, 31, 56, 97, 144, 210, 276, 10, 39, 72, 126, 188, 275, 363, 478, 11, 48, 89, 157, 235, 345, 456, 601, 756, 12, 58, 109, 193, 290, 427, 565, 745, 938, 1164, ...
Links
- 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 13.
- N. J. A. Sloane, Illustration for (m,n) = (2,2), (3,1), (3,2), (3,3) [c_3 = number of triangles, c_4 = number of quadrilaterals; c, e, v = numbers of cells, edges, vertices]
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; ct3 := proc(m,n) local i; global VR; if m=1 or n=1 then max(m,n) else VR(m,n,2)/2+m+n+1; fi; end; # A332354 ct4 := proc(m,n) local i; global VR; if m=1 or n=1 then 0 else VR(m,n,1)/4-VR(m,n,2)/2-m/2-n/2-1; fi; end; # A332356 ct := (m,n) -> ct3(m,n) + ct4(m,n); # A332357 cte := proc(m,n) local i; global VR; if m=1 or n=1 then 2*max(m,n)+1 else VR(m,n,1)/2-VR(m,n,2)/4+m+n; fi; end; # A332359 ctv := (m,n) -> cte(m,n) - ct(m,n) + 1; # A332361 for m from 1 to 12 do lprint([seq(ctv(m,n),n=1..m)]); od: