A259580 Sum of numbers in the n-th antidiagonal of the reciprocity array of 2.

2, 5, 8, 17, 30, 50, 78, 116, 162, 227, 300, 389, 498, 628, 766, 940, 1128, 1347, 1584, 1855, 2146, 2486, 2838, 3236, 3660, 4135, 4626, 5177, 5754, 6392, 7050, 7776, 8524, 9353, 10204, 11127, 12078, 13114, 14170, 15328, 16500, 17775, 19068, 20461, 21900

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..500

Formula

a(n) = Sum_{m=1..n} Sum_{k=0..m-1} floor((n*k + x)/m), where x = 2.

Mathematica

x = 2;  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, A259578, A259579.

Sequence in context: A062318 A034445 A285459 * A316795 A054754 A054755

Adjacent sequences: A259577 A259578 A259579 * A259581 A259582 A259583

Keyword

nonn,easy

Author

Clark Kimberling, Jul 17 2015

Status

approved