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

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

1,8

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 = 1 and [ ] = floor.

Example

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

Mathematica

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 *)

Crossrefs

Cf. A259572, A259576, A259577.

Sequence in context: A172281 A304945 A176298 * A370062 A169623 A245558

Adjacent sequences: A259572 A259573 A259574 * A259576 A259577 A259578

Keyword

nonn,easy,tabl

Author

Clark Kimberling, Jul 01 2015

Status

approved