A259578 Reciprocity array of 2; rectangular, read by antidiagonals.

2, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 4, 3, 4, 2, 2, 4, 5, 5, 4, 2, 2, 5, 6, 6, 6, 5, 2, 2, 5, 6, 8, 8, 6, 5, 2, 2, 6, 8, 10, 10, 10, 8, 6, 2, 2, 6, 9, 11, 12, 12, 11, 9, 6, 2, 2, 7, 9, 12, 14, 15, 14, 12, 9, 7, 2, 2, 7, 11, 14, 16, 17, 17, 16, 14, 11, 7, 2, 2, 8

Offset

1,1

Comments

The "reciprocity law" that Sum{[(n*k+x)/m] : k = 0..m} = Sum{[(m*k+x)/n] : k = 0..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, O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.

Links

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

Formula

T(m,n) = Sum{[(n*k+x)/m] : k = 0..m-1} = Sum{[(m*k+x)/n] : k = 0..n-1}, where x = 2 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.

Northwest corner:

2 2 2 2 2 2 2 2 2 2

2 3 3 4 4 5 5 6 6 7

2 3 3 5 6 6 8 9 9 11

2 4 5 6 8 10 11 12 14 16

2 4 6 8 10 12 14 16 18 20

2 5 6 10 12 15 17 20 21 25

Mathematica

x = 2; s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
TableForm[ Table[s[m, n], {m, 1, 15}, {n, 1, 15}]]
u = Table[s[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten

Crossrefs

Cf. A259572, A259579, A259580.

Sequence in context: A061357 A270966 A138139 * A266547 A127992 A327390

Adjacent sequences: A259575 A259576 A259577 * A259579 A259580 A259581

Keyword

nonn,easy,tabl

Author

Clark Kimberling, Jul 17 2015

Status

approved