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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 18 17:05 EDT 2019. Contains 324214 sequences. (Running on oeis4.)