A259574 Sum of numbers in the n-th antidiagonal of the reciprocity array of 0.

0, 1, 4, 11, 22, 42, 66, 104, 150, 211, 280, 377, 474, 604, 750, 916, 1096, 1323, 1548, 1831, 2122, 2446, 2794, 3212, 3620, 4087, 4590, 5141, 5698, 6360, 6990, 7728, 8484, 9289, 10156, 11091, 12006, 13042, 14122, 15280, 16420, 17727, 18984, 20401, 21852

Offset: 1

Author

Clark Kimberling, Jun 30 2015

Keywords

• nonn
• easy

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, A259573.

Programs

Maple

seq(add(add(floor(n*k/m),k=0..m-1),m=1..n), n=1..100); # Robert Israel, Jul 06 2015

Mathematica

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

PARI

a(n) = {r=0;for(m=1,n,for(k=0,m-1,r=r+floor((n*k)/m)));return(r);} main(size)={return(vector(size,n,a(n)));} /* Anders Hellström, Jul 07 2015 */

Formula

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