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

1,2

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.

Links

Table of n, a(n) for n=1..81.

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

Cf. A249572, A249581, A259583.

Sequence in context: A286245 A279849 A106826 * A139048 A182101 A242289

Adjacent sequences: A259579 A259580 A259581 * A259583 A259584 A259585

Keyword

nonn,easy

Author

Clark Kimberling, Jul 15 2015

Status

approved