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

Author

Clark Kimberling, Jul 15 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..499

Crossrefs

Cf. A259572, A259581, A259582.

Programs

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}]

Formula

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