A259577 Sum of numbers in the n-th antidiagonal of the reciprocity array of 1.

1, 2, 6, 13, 26, 44, 72, 108, 156, 215, 290, 381, 486, 610, 758, 924, 1112, 1329, 1566, 1839, 2134, 2456, 2816, 3220, 3640, 4099, 4608, 5153, 5726, 6368, 7020, 7744, 8504, 9305, 10180, 11103, 12042, 13060, 14146, 15296, 16460, 17739, 19026, 20421, 21876

Offset: 1

Author

Clark Kimberling, Jul 01 2015

Keywords

• nonn

Comments

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.

References

• R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.

Links

• Clark Kimberling, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A259572, A259574, A259575.

Programs

Mathematica

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

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);
main(size)=return(vector(size,n,a(n))) \\ Anders Hellström, Jul 06 2015

PARI

a(n)=sum(m=1,n, sum(k=0,m-1, (n*k+1)\m)) \\ Charles R Greathouse IV, Mar 22 2017

Formula

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

a(n) = n^3 / 4 + O(n^2). - Charles R Greathouse IV, Mar 22 2017