

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.


10



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, 1, 1, 51, 75, 93, 93, 75, 51, 1, 1, 66, 100, 136, 140, 136, 100, 66, 1, 1, 83, 131, 181, 207, 207, 181, 131, 83, 1, 1, 102, 164, 238, 274, 306, 274, 238, 164, 102, 1
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OFFSET

1,5


LINKS



FORMULA

A(n,k) = (1/2) * (f(n,k,1)  f(n,k,2)), where f(n,k,m) = Sum ((nm*x)*(km*y)); n < m*x < n, k < m*y < k, (x,y)=1.


EXAMPLE

Square array begins:
0, 1, 1, 1, 1, ...
1, 6, 11, 18, 27, ...
1, 11, 20, 35, 52, ...
1, 18, 35, 62, 93, ...
1, 27, 52, 93, 140, ...
1, 38, 75, 136, 207, ...


MATHEMATICA

A[n_, k_] := (1/2)(f[n, k, 1]  f[n, k, 2]);
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}];


CROSSREFS

Main diagonal gives A018808. Reading up to the diagonal gives A107348.


KEYWORD



AUTHOR



STATUS

approved



