This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A259581 #12 Mar 02 2025 22:35:21
%S A259581 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 A259581 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 A259581 15,12,12,7,3,3,8,12,15,17,18,18,17,15,12,8,3
%N A259581 Reciprocity array of 3; rectangular, read by antidiagonals.
%C A259581 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). See A259572 for a guide to related sequences.
%D A259581 R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
%H A259581 Clark Kimberling, <a href="/A259581/b259581.txt">Antidiagonals n=1..60, flattened</a>
%F A259581 T(m,n) = Sum_{k=0..m-1} [(n*k+x)/m] = Sum_{k=0..n-1} [(m*k+x)/n], where x = 3 and [ ] = floor.
%F A259581 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 A259581 Northwest corner:
%e A259581 3 3 3 3 3 3 3 3 3 3
%e A259581 3 3 4 4 5 5 6 6 7 7
%e A259581 3 4 6 6 7 9 9 10 12 12
%e A259581 3 4 6 6 9 10 12 12 15 16
%e A259581 3 5 7 9 10 13 15 17 19 20
%e A259581 3 5 9 10 13 15 18 20 24 25
%t A259581 x = 3; s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
%t A259581 TableForm[ Table[s[m, n], {m, 1, 15}, {n, 1, 15}]] (* array *)
%t A259581 u = Table[s[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten (* sequence *)
%Y A259581 Cf. A259572, A259582, A259583.
%K A259581 nonn,easy,tabl
%O A259581 1,1
%A A259581 _Clark Kimberling_, Jul 15 2015