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 A259577 #16 Mar 03 2025 06:47:38
%S A259577 1,2,6,13,26,44,72,108,156,215,290,381,486,610,758,924,1112,1329,1566,
%T A259577 1839,2134,2456,2816,3220,3640,4099,4608,5153,5726,6368,7020,7744,
%U A259577 8504,9305,10180,11103,12042,13060,14146,15296,16460,17739,19026,20421,21876
%N A259577 Sum of numbers in the n-th antidiagonal of the reciprocity array of 1.
%C A259577 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 A259577 R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
%H A259577 Clark Kimberling, <a href="/A259577/b259577.txt">Table of n, a(n) for n = 1..500</a>
%F A259577 a(n) = Sum_{m=1..n} Sum_{k=0..m-1} floor((n*k + x)/m), where x = 1.
%F A259577 a(n) = n^3 / 4 + O(n^2). - _Charles R Greathouse IV_, Mar 22 2017
%t A259577 f[n_] := Sum[Floor[(n*k + 1)/m], {m, n}, {k, 0, m - 1}]; Array[f, 50]
%o A259577 (PARI) a(n)=x=1;r=0;for(m=1,n,for(k=0,m-1,r=r+floor((n*k+x)/m)));return(r);
%o A259577 main(size)=return(vector(size,n,a(n))) \\ _Anders Hellström_, Jul 06 2015
%o A259577 (PARI) a(n)=sum(m=1,n, sum(k=0,m-1, (n*k+1)\m)) \\ _Charles R Greathouse IV_, Mar 22 2017
%Y A259577 Cf. A259572, A259574, A259575.
%K A259577 nonn
%O A259577 1,2
%A A259577 _Clark Kimberling_, Jul 01 2015