A259577 - OEIS

%I #12 Mar 22 2017 17:13:45

%S 1,2,6,13,26,44,72,108,156,215,290,381,486,610,758,924,1112,1329,1566,

%T 1839,2134,2456,2816,3220,3640,4099,4608,5153,5726,6368,7020,7744,

%U 8504,9305,10180,11103,12042,13060,14146,15296,16460,17739,19026,20421,21876

%N Sum of numbers in the n-th antidiagonal of the reciprocity array of 1.

%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="/A259577/b259577.txt">Table of n, a(n) for n = 1..500</a>

%F a(n) = sum{sum{floor((n*k + x)/m), k=0..m-1, m=1..n}, where x = 1.

%F a(n) = n^3 / 4 + O(n^2). - _Charles R Greathouse IV_, Mar 22 2017

%t f[n_] := Sum[Floor[(n*k + 1)/m], {m, n}, {k, 0, m - 1}]; Array[f, 50]

%o (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 main(size)=return(vector(size,n,a(n))) \\ _Anders Hellström_, Jul 06 2015

%o (PARI) a(n)=sum(m=1,n, sum(k=0,m-1, (n*k+1)\m)) \\ _Charles R Greathouse IV_, Mar 22 2017

%Y Cf. A259572, A259574, A259575.

%K nonn

%O 1,2

%A _Clark Kimberling_, Jul 01 2015