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 A259575 #15 Mar 02 2025 22:35:29
%S A259575 1,1,1,1,1,1,1,2,2,1,1,2,3,2,1,1,3,4,4,3,1,1,3,5,6,5,3,1,1,4,6,7,7,6,
%T A259575 4,1,1,4,7,8,10,8,7,4,1,1,5,8,10,11,11,10,8,5,1,1,5,9,12,13,15,13,12,
%U A259575 9,5,1,1,6,10,13,15,16,16,15,13,10,6,1,1,6
%N A259575 Reciprocity array of 1; rectangular, read by antidiagonals.
%C A259575 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 A259575 R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
%H A259575 Clark Kimberling, <a href="/A259575/b259575.txt">Antidiagonals n=1..60, flattened</a>
%F A259575 T(m,n) = Sum_{k=0..m-1} [(n*k+x)/m] = Sum_{k=0..n-1} [(m*k+x)/n], where x = 1 and [ ] = floor.
%e A259575 Northwest corner:
%e A259575 1 1 1 1 1 1 1 1 1 1
%e A259575 1 1 2 2 3 3 4 4 5 5
%e A259575 1 2 3 4 5 6 7 8 9 10
%e A259575 1 2 4 6 7 8 10 12 13 14
%e A259575 1 3 5 7 10 11 13 15 17 20
%e A259575 1 3 6 8 11 15 16 18 21 23
%e A259575 1 4 7 10 13 16 21 22 25 28
%t A259575 x = 1; s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
%t A259575 TableForm[ Table[s[m, n], {m, 1, 15}, {n, 1, 15}]] (* array *)
%t A259575 Table[s[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten (* sequence *)
%Y A259575 Cf. A259572, A259576, A259577.
%K A259575 nonn,easy,tabl
%O A259575 1,8
%A A259575 _Clark Kimberling_, Jul 01 2015