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%I #6 Jul 19 2015 11:01:52
%S 1,2,3,2,1,4,3,4,5,4,3,6,3,4,5,6,3,6,3,6,7,6,3,10,3,6,7,8,3,12,3,8,9,
%T 6,5,12,3,6,9,10,3,12,3,10,9,6,3,16,5,8,9,10,3,10,5,10,9,6,3,20,3,6,9,
%U 10,5,14,3,10,9,12,3,16,3,6,11,10,9,14,3,14
%N Number of distinct differences in row n of the reciprocity array of 2.
%C The "reciprocity law" that Sum{[(n*k+x)/m] : k = 0..m} = Sum{[(m*k+x)/n] : k = 0..n} where x is a real number and m and n are positive integers, is proved in Section 3.5 of Concrete Mathematics (see References). See A259572 for a guide to related sequences.
%D R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
%e In the array at A259578, row 6 is (2,5,6,10,12,15,17,20,21,25,27,...), with differences (3,1,4,2,3,2,3,1,4,2,...), and distinct differences {1,2,3,4}, so that a(4) = 4.
%t x = 2; s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
%t t[m_] := Table[s[m, n], {n, 1, 1000}];
%t u = Table[Length[Union[Differences[t[m]]]], {m, 1, 120}]
%Y Cf. A249572, A249577, A259580.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Jul 17 2015
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