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%I #4 Jul 15 2015 17:32:01
%S 3,3,3,3,3,3,3,4,4,3,3,4,6,4,3,3,5,6,6,5,3,3,5,7,6,7,5,3,3,6,9,9,9,9,
%T 6,3,3,6,9,10,10,10,9,6,3,3,7,10,12,13,13,12,10,7,3,3,7,12,12,15,15,
%U 15,12,12,7,3,3,8,12,15,17,18,18,17,15,12,8,3
%N Reciprocity array of 3; 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="/A259581/b259581.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 = 3 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.
%e Northwest corner:
%e 3 3 3 3 3 3 3 3 3 3
%e 3 3 4 4 5 5 6 6 7 7
%e 3 4 6 6 7 9 9 10 12 12
%e 3 5 7 9 10 12 12 15 16 18
%e 3 5 9 10 13 15 17 19 20 23
%t x = 3; 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. A259572, A259582, A259583.
%K nonn,easy,tabl
%O 1,1
%A _Clark Kimberling_, Jul 15 2015
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