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%I #18 Feb 12 2020 01:05:59
%S 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,
%T 144,210,276,10,39,72,126,188,275,363,478,11,48,89,157,235,345,456,
%U 601,756,12,58,109,193,290,427,565,745,938,1164,13,69,130,231,347,511,675,890,1120,1390,1660
%N 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.
%H M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. <a href="https://doi.org/10.1137/140978090">On the minimal teaching sets of two-dimensional threshold functions</a>. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Theorem 13.
%H N. J. A. Sloane, <a href="/A332357/a332357.pdf">Illustration for (m,n) = (2,2), (3,1), (3,2), (3,3)</a> [c_3 = number of triangles, c_4 = number of quadrilaterals; c, e, v = numbers of cells, edges, vertices]
%F T(m,n) = A332359(m,n) - A332357(m,n) + 1 (Euler's formula).
%e Triangle begins:
%e 3,
%e 4, 6,
%e 5, 9, 14,
%e 6, 13, 22, 36,
%e 7, 18, 31, 52, 76,
%e 8, 24, 43, 74, 110, 160,
%e 9, 31, 56, 97, 144, 210, 276,
%e 10, 39, 72, 126, 188, 275, 363, 478,
%e 11, 48, 89, 157, 235, 345, 456, 601, 756,
%e 12, 58, 109, 193, 290, 427, 565, 745, 938, 1164,
%e ...
%p VR := proc(m,n,q) local a,i,j; a:=0;
%p for i from -m+1 to m-1 do for j from -n+1 to n-1 do
%p if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
%p ct3 := proc(m,n) local i; global VR;
%p if m=1 or n=1 then max(m,n) else VR(m,n,2)/2+m+n+1; fi; end; # A332354
%p ct4 := proc(m,n) local i; global VR;
%p 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
%p ct := (m,n) -> ct3(m,n) + ct4(m,n); # A332357
%p cte := proc(m,n) local i; global VR;
%p 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
%p ctv := (m,n) -> cte(m,n) - ct(m,n) + 1; # A332361
%p for m from 1 to 12 do lprint([seq(ctv(m,n),n=1..m)]); od:
%Y Cf. A332350-A331360.
%Y Main diagonal is A332362.
%K nonn,tabl
%O 1,1
%A _N. J. A. Sloane_, Feb 11 2020
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