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

1,1

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.

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

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.

Example

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

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 *)

Crossrefs

Cf. A259572, A259582, A259583.

Sequence in context: A162844 A115787 A300959 * A105592 A210745 A187471

Adjacent sequences: A259578 A259579 A259580 * A259582 A259583 A259584

Keyword

nonn,easy,tabl,changed

Author

Clark Kimberling, Jul 15 2015

Status

approved