A259581 Reciprocity array of 3; rectangular, read by antidiagonals.

3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 4, 6, 4, 3, 3, 5, 6, 6, 5, 3, 3, 5, 7, 6, 7, 5, 3, 3, 6, 9, 9, 9, 9, 6, 3, 3, 6, 9, 10, 10, 10, 9, 6, 3, 3, 7, 10, 12, 13, 13, 12, 10, 7, 3, 3, 7, 12, 12, 15, 15, 15, 12, 12, 7, 3, 3, 8, 12, 15, 17, 18, 18, 17, 15, 12, 8, 3

Offset: 1

Author

Clark Kimberling, Jul 15 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). See A259572 for a guide to related sequences.

Examples

			Northwest corner:
  3   3   3   3   3   3   3   3   3   3
  3   3   4   4   5   5   6   6   7   7
  3   4   6   6   7   9   9   10  12  12
  3   4   6   6   9   10  12  12  15  16
  3   5   7   9   10  13  15  17  19  20
  3   5   9   10  13  15  18  20  24  25
		

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. A259572, A259582, A259583.

Programs

Mathematica

x = 3;  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 = 3 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.