A259583 Sum of numbers in the n-th antidiagonal of the reciprocity array of 3.

3, 6, 13, 19, 34, 55, 84, 120, 174, 231, 310, 399, 510, 634, 786, 948, 1144, 1359, 1602, 1863, 2176, 2496, 2860, 3256, 3680, 4147, 4662, 5189, 5782, 6412, 7080, 7792, 8574, 9369, 10228, 11151, 12114, 13132, 14230, 15344, 16540, 17805, 19110, 20481, 21948

Offset

1,1

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..499

Formula

a(n) = Sum_{m=1..n} Sum_{k=0..m-1} floor((n*k + x)/m), where x = 3.

Mathematica

x = 3;  v[n_] := Sum[Sum[Floor[(n*k + x)/m], {k, 0, m - 1}], {m, 1, n}];
Table[v[n], {n, 1, 120}]

Crossrefs

Cf. A259572, A259581, A259582.

Sequence in context: A137042 A080546 A080554 * A064349 A101965 A230370

Adjacent sequences: A259580 A259581 A259582 * A259584 A259585 A259586

Keyword

nonn,easy,changed

Author

Clark Kimberling, Jul 15 2015

Status

approved