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

1,3

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

Cf. A259572, A259573.

Sequence in context: A177853 A008016 A295957 * A008249 A174405 A016274

Adjacent sequences: A259571 A259572 A259573 * A259575 A259576 A259577

Keyword

nonn,easy

Author

Clark Kimberling, Jun 30 2015

Status

approved