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A259582
Number of distinct differences in row n of the reciprocity array of 3.
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
1,2
COMMENTS
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,
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, O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
EXAMPLE
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.
MATHEMATICA
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}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 15 2015
STATUS
approved