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A259572 Reciprocity array of 0; rectangular, read by antidiagonals.

%I A259572 #40 Mar 03 2025 06:48:03
%S A259572 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 A259572 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 A259572 5,0,0,5,9,12,14,15,15,14,12,9,5,0,0,6,10
%N A259572 Reciprocity array of 0; rectangular, read by antidiagonals.
%C A259572 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). For every x, the reciprocity array is symmetric, and the principal diagonal consists primarily of triangular numbers, A000217.
%C A259572 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 A259572    x              array     differences     sums
%C A259572    0             A259572      A259573      A259574
%C A259572    1             A259575      A259576      A259577
%C A259572    2             A259578      A259579      A259580
%C A259572    3             A259581      A259582      A259583
%D A259572 R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
%H A259572 Clark Kimberling, <a href="/A259572/b259572.txt">Antidiagonals n=1..60, flattened</a>
%F A259572 T(m,n) = Sum_{k=0..m-1} [(n*k+x)/m] = Sum_{k=0..n-1} [(m*k+x)/n], where x = 0 and [ ] = floor.
%F A259572 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 A259572 T(m,n) = (m*n - m - n + gcd(m,n))/2. - _Witold Dlugosz_, Apr 07 2021
%e A259572 Northwest corner:
%e A259572   0   0   0   0   0   0   0   0   0   0
%e A259572   0   1   1   2   2   3   3   4   4   5
%e A259572   0   1   3   3   4   6   6   7   9   9
%e A259572   0   2   3   6   6   8   9  12  12  14
%e A259572   0   2   4   6  10  10  12  14  16  20
%e A259572   0   3   6   8  10  15  15  18  21  23
%t A259572 x = 0;  s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
%t A259572 TableForm[ Table[s[m, n], {m, 1, 15}, {n, 1, 15}]] (* array *)
%t A259572 u = Table[s[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten (* sequence *)
%Y A259572 Cf. A259573, A259574.
%K A259572 nonn,easy,tabl
%O A259572 1,12
%A A259572 _Clark Kimberling_, Jun 30 2015