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A295707 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of lines through at least 2 points of an n X k grid of points.

%I A295707 #35 Jun 05 2023 09:19:43
%S A295707 0,1,1,1,6,1,1,11,11,1,1,18,20,18,1,1,27,35,35,27,1,1,38,52,62,52,38,
%T A295707 1,1,51,75,93,93,75,51,1,1,66,100,136,140,136,100,66,1,1,83,131,181,
%U A295707 207,207,181,131,83,1,1,102,164,238,274,306,274,238,164,102,1
%N A295707 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of lines through at least 2 points of an n X k grid of points.
%H A295707 Seiichi Manyama, <a href="/A295707/b295707.txt">Antidiagonals n = 1..140, flattened</a>
%H A295707 Seppo Mustonen, <a href="http://www.survo.fi/papers/PointsInGrid.pdf">On lines and their intersection points in a rectangular grid of points</a>
%H A295707 Seppo Mustonen, <a href="/A018808/a018808.pdf">On lines and their intersection points in a rectangular grid of points</a> [Local copy]
%F A295707 A(n,k) = (1/2) * (f(n,k,1) - f(n,k,2)), where f(n,k,m) = Sum ((n-|m*x|)*(k-|m*y|)); -n < m*x < n, -k < m*y < k, (x,y)=1.
%e A295707 Square array begins:
%e A295707    0,  1,  1,   1,   1, ...
%e A295707    1,  6, 11,  18,  27, ...
%e A295707    1, 11, 20,  35,  52, ...
%e A295707    1, 18, 35,  62,  93, ...
%e A295707    1, 27, 52,  93, 140, ...
%e A295707    1, 38, 75, 136, 207, ...
%t A295707 A[n_, k_] := (1/2)(f[n, k, 1] - f[n, k, 2]);
%t A295707 f[n_, k_, m_] := Sum[If[GCD[mx/m, my/m] == 1, (n - Abs[mx])(k - Abs[my]), 0], {mx, -n, n}, {my, -k, k}];
%t A295707 Table[A[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 04 2023 *)
%Y A295707 Columns k=2..10 give A160842, A160843, A160844, A160845, A160846, A160847, A160848, A160849, A160850.
%Y A295707 Main diagonal gives A018808. Reading up to the diagonal gives A107348.
%K A295707 nonn,tabl
%O A295707 1,5
%A A295707 _Seiichi Manyama_, Nov 26 2017