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

Author

Clark Kimberling, Jun 30 2015

Keywords

• nonn
• easy
• tabl

Comments

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

Examples

			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
		

References

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

Links

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

Crossrefs

Cf. A259573, A259574.

Programs

Mathematica

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 *)

Formula

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