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

Author

Clark Kimberling, Jul 17 2015

Keywords

• nonn
• easy

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

Crossrefs

Cf. A259572, A259578, A259579.

Programs

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}]

Formula

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