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%I #29 Apr 08 2021 07:24:46
%S 0,0,0,0,1,0,0,1,1,0,0,2,3,2,0,0,2,3,3,2,0,0,3,4,6,4,3,0,0,3,6,6,6,6,
%T 3,0,0,4,6,8,10,8,6,4,0,0,4,7,9,10,10,9,7,4,0,0,5,9,12,12,15,12,12,9,
%U 5,0,0,5,9,12,14,15,15,14,12,9,5,0,0,6,10
%N Reciprocity array of 0; rectangular, read by antidiagonals.
%C 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.
%C 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.
%C x array differences sums
%C 0 A259572 A259573 A259574
%C 1 A259575 A259576 A259577
%C 2 A259578 A259579 A249580
%C 3 A259581 A259582 A249583
%D R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
%H Clark Kimberling, <a href="/A259572/b259572.txt">Antidiagonals n=1..60, flattened </a>
%F 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.
%F 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.
%F T(m,n) = (m*n - m - n + gcd(m,n))/2. - _Witold Dlugosz_, Apr 07 2021
%e Northwest corner:
%e 0 0 0 0 0 0 0 0 0 0
%e 0 1 1 2 2 3 3 4 4 5
%e 0 1 3 3 4 6 6 7 9 9
%e 0 2 3 6 6 8 9 12 12 14
%e 0 2 4 6 10 10 12 14 16 20
%e 0 3 6 8 10 15 15 18 21 23
%t x = 0; s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
%t TableForm[ Table[s[m, n], {m, 1, 15}, {n, 1, 15}]] (* array *)
%t u = Table[s[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten (* sequence *)
%Y Cf. A259573, A259574.
%K nonn,easy,tabl
%O 1,12
%A _Clark Kimberling_, Jun 30 2015
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