OFFSET
1,12
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). For every x, the reciprocity array is symmetric, and the principal diagonal consists primarily of triangular numbers, A000217.
In the following guide, the sequence in column 3 is the number of distinct terms in the difference sequence of row n of the reciprocity array of x; sequence in column 4 is the sum of numbers in the n-th antidiagonal of the array.
x array differences sums
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.
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 in this case.
T(m,n) = (m*n - m - n + gcd(m,n))/2. - Witold Dlugosz, Apr 07 2021
EXAMPLE
Northwest corner:
0 0 0 0 0 0 0 0 0 0
0 1 1 2 2 3 3 4 4 5
0 1 3 3 4 6 6 7 9 9
0 2 3 6 6 8 9 12 12 14
0 2 4 6 10 10 12 14 16 20
0 3 6 8 10 15 15 18 21 23
MATHEMATICA
x = 0; 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
KEYWORD
AUTHOR
Clark Kimberling, Jun 30 2015
STATUS
approved