A259574 -id:A259574 - cp's OEIS frontend

A259572 Reciprocity array of 0; rectangular, read by antidiagonals.

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 3, 2, 0, 0, 2, 3, 3, 2, 0, 0, 3, 4, 6, 4, 3, 0, 0, 3, 6, 6, 6, 6, 3, 0, 0, 4, 6, 8, 10, 8, 6, 4, 0, 0, 4, 7, 9, 10, 10, 9, 7, 4, 0, 0, 5, 9, 12, 12, 15, 12, 12, 9, 5, 0, 0, 5, 9, 12, 14, 15, 15, 14, 12, 9, 5, 0, 0, 6, 10

Offset: 1

Clark Kimberling, Jun 30 2015

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). For every x, the reciprocity array is symmetric, and the principal diagonal consists primarily of triangular numbers, A000217.
In the following guide, the sequence in column 3 is the number of distinct terms in the difference sequence of row n of the reciprocity array of x; sequence in column 4 is the sum of numbers in the n-th antidiagonal of the array.
   x              array     differences     sums
   0             A259572      A259573      A259574
   1             A259575      A259576      A259577
   2             A259578      A259579      A259580
   3             A259581      A259582      A259583

			Northwest corner:
  0   0   0   0   0   0   0   0   0   0
  0   1   1   2   2   3   3   4   4   5
  0   1   3   3   4   6   6   7   9   9
  0   2   3   6   6   8   9  12  12  14
  0   2   4   6  10  10  12  14  16  20
  0   3   6   8  10  15  15  18  21  23

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

Clark Kimberling, Antidiagonals n=1..60, flattened

Cf. A259573, A259574.

x = 0;  s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
TableForm[ Table[s[m, n], {m, 1, 15}, {n, 1, 15}]] (* array *)
u = Table[s[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten (* sequence *)

T(m,n) = Sum_{k=0..m-1} [(n*k+x)/m] = Sum_{k=0..n-1} [(m*k+x)/n], where x = 0 and [ ] = floor.
Note that if [x] = [y], then [(n*k+x)/m] = [(n*k+y)/m], so that the reciprocity arrays for x and y are identical in this case.
T(m,n) = (m*n - m - n + gcd(m,n))/2. - Witold Dlugosz, Apr 07 2021

A259573 Number of distinct differences in row n of the reciprocity array of 0.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jun 30 2015

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).

			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

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

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

Cf. A249572, A259574.

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 *)

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

A259577 Sum of numbers in the n-th antidiagonal of the reciprocity array of 1.

Original entry on oeis.org

1, 2, 6, 13, 26, 44, 72, 108, 156, 215, 290, 381, 486, 610, 758, 924, 1112, 1329, 1566, 1839, 2134, 2456, 2816, 3220, 3640, 4099, 4608, 5153, 5726, 6368, 7020, 7744, 8504, 9305, 10180, 11103, 12042, 13060, 14146, 15296, 16460, 17739, 19026, 20421, 21876

Offset: 1

Clark Kimberling, Jul 01 2015

nonn

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

Clark Kimberling, Table of n, a(n) for n = 1..500

Cf. A259572, A259574, A259575.

f[n_] := Sum[Floor[(n*k + 1)/m], {m, n}, {k, 0, m - 1}]; Array[f, 50]

a(n)=x=1;r=0;for(m=1,n,for(k=0,m-1,r=r+floor((n*k+x)/m)));return(r);
main(size)=return(vector(size,n,a(n))) \\ Anders Hellström, Jul 06 2015

a(n)=sum(m=1,n, sum(k=0,m-1, (n*k+1)\m)) \\ Charles R Greathouse IV, Mar 22 2017

a(n) = Sum_{m=1..n} Sum_{k=0..m-1} floor((n*k + x)/m), where x = 1.

a(n) = n^3 / 4 + O(n^2). - Charles R Greathouse IV, Mar 22 2017

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A259572 Reciprocity array of 0; rectangular, read by antidiagonals.

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A259573 Number of distinct differences in row n of the reciprocity array of 0.

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A259577 Sum of numbers in the n-th antidiagonal of the reciprocity array of 1.

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