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A259573 Number of distinct differences in row n of the reciprocity array of 0. 3
1, 2, 3, 4, 3, 4, 3, 6, 5, 6, 3, 8, 3, 6, 7, 8, 3, 8, 3, 8, 9, 6, 3, 12, 5, 6, 7, 10, 3, 14, 3, 10, 9, 6, 9, 14, 3, 6, 9, 12, 3, 12, 3, 12, 11, 6, 3, 18, 5, 10, 9, 12, 3, 12, 9, 14, 9, 6, 3, 22, 3, 6, 13, 12, 9, 14, 3, 12, 9, 14, 3, 18, 3, 6, 13, 12, 9, 16 (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).

REFERENCES

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

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

EXAMPLE

In the array at A259572, row 4 is (0,2,3,6,6,8,9,12,12,14,15,...), with differences (2,1,3,0,2,1,3,0,2,1,3,0, ...), and distinct differences {0,1,2,3}, so that a(4) = 4. Example corrected by Antti Karttunen, Nov 30 2021

MATHEMATICA

x = 0;  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}]  (* A259573 *)

PROG

(PARI)

A259572(m, n) = ((m*n - m - n + gcd(m, n))/2); \\ After Witold Dlugosz's formula for A259572.

A259573(n) = #Set(vector(n, k, A259572(n, 1+k)-A259572(n, k))); \\ Antti Karttunen, Nov 30 2021

CROSSREFS

Cf. A249572, A259574.

Sequence in context: A304730 A323374 A324197 * A108015 A344322 A030398

Adjacent sequences:  A259570 A259571 A259572 * A259574 A259575 A259576

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jun 30 2015

STATUS

approved

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Last modified May 16 18:07 EDT 2022. Contains 353719 sequences. (Running on oeis4.)