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Array T read by antidiagonals; T(i,j)=number of nonnegative slopes of lines determined by two points in the triangular set {(x,y): 0<=x<=i, 0<=y<=j-j*x/i} of lattice points.
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%I #13 Jun 10 2017 08:06:53

%S 0,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,3,4,3,1,1,1,1,3,4,4,3,

%T 1,1,1,1,4,5,6,5,4,1,1,1,1,4,6,6,6,6,4,1,1,1,1,5,6,8,10,8,6,5,1,1,1,1,

%U 5,7,9,10,10,9,7,5,1,1,1,1,6,9,11,11,12,11,11

%N Array T read by antidiagonals; T(i,j)=number of nonnegative slopes of lines determined by two points in the triangular set {(x,y): 0<=x<=i, 0<=y<=j-j*x/i} of lattice points.

%e The array begins:

%e 0 1 1 1 1 1 1 1 1...

%e 1 1 1 1 1 1 1 1 1...

%e 1 1 2 2 3 3 4 4 5...

%e 1 1 2 4 4 5 6 6 7...

%e 1 1 3 4 6 6 8 9 11...

%e ...

%t t[i_,j_] := If[i==0||j==0, 1-KroneckerDelta[i+j], 1+Length[Union[Divide@@#& /@ Select[-Subtract@@@Subsets[Flatten[Table[{x,y}, {x,0,i}, {y,0,j-j*x/i}], 1], {2}], And@@Positive/@#&]]]];

%t (*Table[t[i,j], {i,0,10}, {j,0,10}]//TableForm*)

%t Flatten@Table[t[j,i-j], {i,0,20}, {j,0,i}]

%t (* _Andrey Zabolotskiy_, Jun 09 2017 *)

%Y Cf. A049600, A049615, A049639, A049687, A049695, A049702. Cf. A049706-A049709. Main diagonal: A002088.

%K nonn,tabl

%O 0,13

%A _Clark Kimberling_

%E Name corrected by _Michel Marcus_ and _Andrey Zabolotskiy_, Jun 10 2017