OFFSET
1,3
COMMENTS
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 A259575 for a guide to related sequences.
REFERENCES
R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..500
FORMULA
a(n) = Sum{m=1..n, Sum(k=0..m-1, floor(n*k/m)).
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]
PROG
(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 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 30 2015
STATUS
approved