A259576 -id:A259576 - 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

A259575 Reciprocity array of 1; rectangular, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 4, 4, 3, 1, 1, 3, 5, 6, 5, 3, 1, 1, 4, 6, 7, 7, 6, 4, 1, 1, 4, 7, 8, 10, 8, 7, 4, 1, 1, 5, 8, 10, 11, 11, 10, 8, 5, 1, 1, 5, 9, 12, 13, 15, 13, 12, 9, 5, 1, 1, 6, 10, 13, 15, 16, 16, 15, 13, 10, 6, 1, 1, 6

Offset: 1

Clark Kimberling, Jul 01 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). See A259572 for a guide to related sequences.

			Northwest corner:
  1   1   1   1   1   1   1   1   1   1
  1   1   2   2   3   3   4   4   5   5
  1   2   3   4   5   6   7   8   9   10
  1   2   4   6   7   8   10  12  13  14
  1   3   5   7   10  11  13  15  17  20
  1   3   6   8   11  15  16  18  21  23
  1   4   7   10  13  16  21  22  25  28

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. A259572, A259576, A259577.

x = 1;  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 *)
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 = 1 and [ ] = floor.

cp's OEIS Frontend

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

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