A249581 -id:A249581 - cp's OEIS frontend

A259582 Number of distinct differences in row n of the reciprocity array of 3.

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, 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, 9, 14, 3, 12, 9, 12, 3, 14, 3, 6, 13, 12, 5, 14, 3, 14, 7

Offset: 1

Clark Kimberling, Jul 15 2015

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.

			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.

R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.

Cf. A249572, A249581, A259583.

x = 3; s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
t[m_] := Table[s[m, n], {n, 1, 1000}];
Table[Length[Union[Differences[t[m]]]], {m, 1, 120}]

cp's OEIS Frontend

A259582 Number of distinct differences in row n of the reciprocity array of 3.

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