OFFSET
1,8
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).
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 = 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 14 15
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
KEYWORD
AUTHOR
Clark Kimberling, Jul 01 2015
STATUS
approved