OFFSET
1,2
COMMENTS
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.
REFERENCES
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
EXAMPLE
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(6) = 4.
MATHEMATICA
x = 2; s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
t[m_] := Table[s[m, n], {n, 1, 1000}];
u = Table[Length[Union[Differences[t[m]]]], {m, 1, 120}]
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Clark Kimberling, Jul 17 2015
STATUS
approved