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%I #6 Jul 19 2015 11:01:36
%S 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,
%T 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,
%U 12,9,7,2,2,7,11,14,16,17,17,16,14,11,7,2,2,8
%N Reciprocity array of 2; 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). See A259572 for a guide to related sequences.
%D R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
%H Clark Kimberling, <a href="/A259578/b259578.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 = 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.
%F Northwest corner:
%F 2 2 2 2 2 2 2 2 2 2
%F 2 3 3 4 4 5 5 6 6 7
%F 2 3 3 5 6 6 8 9 9 11
%F 2 4 5 6 8 10 11 12 14 16
%F 2 4 6 8 10 12 14 16 18 20
%F 2 5 6 10 12 15 17 20 21 25
%t x = 2; 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}]]
%t u = Table[s[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten
%Y Cf. A259572, A259579, A259580.
%K nonn,easy,tabl
%O 1,1
%A _Clark Kimberling_, Jul 17 2015
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