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%I #4 Jul 15 2015 17:32:23
%S 1,2,3,4,3,4,1,4,3,4,3,6,3,4,7,6,3,6,3,6,5,6,3,8,5,6,5,8,3,8,3,8,7,6,
%T 5,10,3,6,9,8,3,12,3,12,9,6,3,14,3,8,9,12,3,10,9,10,9,6,3,18,3,6,7,10,
%U 9,14,3,12,9,12,3,14,3,6,13,12,5,14,3,14,7
%N Number of distinct differences in row n of the reciprocity array of 3.
%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,
%C 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 A259581, row 4 is (3,4,6,6,9,10,12,12,15,16,...), with differences (1,2,0,3,1,2,2,3,1,...),
%e and distinct differences {0,1,2,3}, so that a(4) = 4.
%t x = 3; s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
%t t[m_] := Table[s[m, n], {n, 1, 1000}];
%t Table[Length[Union[Differences[t[m]]]], {m, 1, 120}]
%Y Cf. A249572, A249581, A259583.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Jul 15 2015
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