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 A259583 #8 Mar 03 2025 06:48:20
%S A259583 3,6,13,19,34,55,84,120,174,231,310,399,510,634,786,948,1144,1359,
%T A259583 1602,1863,2176,2496,2860,3256,3680,4147,4662,5189,5782,6412,7080,
%U A259583 7792,8574,9369,10228,11151,12114,13132,14230,15344,16540,17805,19110,20481,21948
%N A259583 Sum of numbers in the n-th antidiagonal of the reciprocity array of 3.
%C A259583 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 A259583 R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
%H A259583 Clark Kimberling, <a href="/A259583/b259583.txt">Table of n, a(n) for n = 1..499</a>
%F A259583 a(n) = Sum_{m=1..n} Sum_{k=0..m-1} floor((n*k + x)/m), where x = 3.
%t A259583 x = 3; v[n_] := Sum[Sum[Floor[(n*k + x)/m], {k, 0, m - 1}], {m, 1, n}];
%t A259583 Table[v[n], {n, 1, 120}]
%Y A259583 Cf. A259572, A259581, A259582.
%K A259583 nonn,easy
%O A259583 1,1
%A A259583 _Clark Kimberling_, Jul 15 2015