This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A259582 #8 Mar 03 2025 06:47:47
%S A259582 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 A259582 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 A259582 9,14,3,12,9,12,3,14,3,6,13,12,5,14,3,14,7
%N A259582 Number of distinct differences in row n of the reciprocity array of 3.
%C A259582 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). See A259572 for a guide to related sequences.
%D A259582 R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
%e A259582 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,...), and distinct differences {0,1,2,3}, so that a(4) = 4.
%t A259582 x = 3; s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
%t A259582 t[m_] := Table[s[m, n], {n, 1, 1000}];
%t A259582 Table[Length[Union[Differences[t[m]]]], {m, 1, 120}]
%Y A259582 Cf. A249572, A249581, A259583.
%K A259582 nonn,easy
%O A259582 1,2
%A A259582 _Clark Kimberling_, Jul 15 2015