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A259573 Number of distinct differences in row n of the reciprocity array of 0.

%I A259573 #14 Mar 03 2025 06:47:28
%S A259573 1,2,3,4,3,4,3,6,5,6,3,8,3,6,7,8,3,8,3,8,9,6,3,12,5,6,7,10,3,14,3,10,
%T A259573 9,6,9,14,3,6,9,12,3,12,3,12,11,6,3,18,5,10,9,12,3,12,9,14,9,6,3,22,3,
%U A259573 6,13,12,9,14,3,12,9,14,3,18,3,6,13,12,9,16
%N A259573 Number of distinct differences in row n of the reciprocity array of 0.
%C A259573 The "reciprocity law" that Sum_{k=0..m} [(n*k+x)/m] = Sum_{k=0..n} [(m*k+x)/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).
%D A259573 R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
%H A259573 Antti Karttunen, <a href="/A259573/b259573.txt">Table of n, a(n) for n = 1..20000</a>
%e A259573 In the array at A259572, row 4 is (0,2,3,6,6,8,9,12,12,14,15,...), with differences (2,1,3,0,2,1,3,0,2,1,3,0, ...), and distinct differences {0,1,2,3}, so that a(4) = 4. Example corrected by _Antti Karttunen_, Nov 30 2021
%t A259573 x = 0;  s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
%t A259573 t[m_] := Table[s[m, n], {n, 1, 1000}];
%t A259573 u = Table[Length[Union[Differences[t[m]]]], {m, 1, 120}]  (* A259573 *)
%o A259573 (PARI)
%o A259573 A259572(m,n) = ((m*n - m - n + gcd(m,n))/2); \\ After _Witold Dlugosz_'s formula for A259572.
%o A259573 A259573(n) = #Set(vector(n,k,A259572(n,1+k)-A259572(n,k))); \\ _Antti Karttunen_, Nov 30 2021
%Y A259573 Cf. A249572, A259574.
%K A259573 nonn,easy
%O A259573 1,2
%A A259573 _Clark Kimberling_, Jun 30 2015