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

1,2

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 A259572 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{sum{floor((n*k + x)/m), k=0..m-1, m=1..n}, where x = 1.

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A259572, A259574, A259575.

Sequence in context: A335439 A181522 A031872 * A172348 A254821 A192953

Adjacent sequences: A259574 A259575 A259576 * A259578 A259579 A259580

Keyword

nonn

Author

Clark Kimberling, Jul 01 2015

Status

approved