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Triangle read by rows: T(m,n) = number of threshold functions (the function u_{0,2}(m,n) of Alekseyev et al. 2015) for m >= n >= 2.
1

%I #13 Feb 13 2020 00:58:53

%S 3,6,13,9,21,33,12,30,49,73,15,40,66,99,133,18,51,85,130,177,237,21,

%T 63,106,164,224,301,381,24,76,130,202,277,374,475,593,27,90,154,241,

%U 331,448,570,713,857,30,105,182,287,395,538,687,862,1039,1261,33,121,211,335,462,632,808,1016,1226,1489,1757

%N Triangle read by rows: T(m,n) = number of threshold functions (the function u_{0,2}(m,n) of Alekseyev et al. 2015) for m >= n >= 2.

%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 11.

%e Triangle begins:

%e 3,

%e 6, 13,

%e 9, 21, 33,

%e 12, 30, 49, 73,

%e 15, 40, 66, 99, 133,

%e 18, 51, 85, 130, 177, 237,

%e 21, 63, 106, 164, 224, 301, 381,

%e 24, 76, 130, 202, 277, 374, 475, 593,

%e 27, 90, 154, 241, 331, 448, 570, 713, 857,

%e ...

%p VQ := proc(m,n,q) local eps,a,i,j; eps := 10^(-6); a:=0;

%p for i from ceil(-m+eps) to floor(m-eps) do

%p for j from ceil(-n+eps) to floor(n-eps) do

%p if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;

%p VS := proc(m,n) local a,i,j; a:=0;

%p for i from 1 to m-1 do for j from 1 to n-1 do

%p if gcd(i,j)=1 then a:=a+1; fi; od: od: a; end; # A331781

%p u02:=(m,n) -> VQ(m,n,2)+2-2*VQ(m/2,n/2,1)+VS(m,n); # This sequence

%p for m from 2 to 12 do lprint([seq(u02(m,n),n=2..m)]); od:

%Y Cf. A332350, A332352, A332363, A331781.

%Y Main diagonal is A332366.

%K nonn,tabl

%O 2,1

%A _N. J. A. Sloane_, Feb 11 2020