OFFSET
1,6
COMMENTS
For the denominators see A264389.
This gives the numerators of the rational numbers r(n) = s(1,n), where s(h,k) = Sum_{r=1..(k-1)} (r/k)*(h*r/k - floor(h*r/k)- 1/2), k >=1, are the Dedekind sums. See the Apostol reference, pp. 52, 61-69, 72-73, and the Weisstein link, where GCD(h,k) = 1 is assumed.
s(h,k) = Sum_{r = 1..k} ((r/k))*((h*r/k)) with ((x)) = x - floor(x) - 1/2 if x is not an integer, else 0.
s(h,k) = (Sum_{r=1..(k-1)} cot(Pi*h*r/k)*cot(Pi*r/k))/(4*k), k >= 1, r and h integers. Exercise 11, p. 72 of the Apostol reference.
6*n*s(1,n) = binomial(n-1, 2) = A161680(n-1), n >= 1.
REFERENCES
Apostol, Tom, M., Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990.
LINKS
Eric Weisstein's World of Mathematics, Dedekind Sum.
FORMULA
a(n) = numerator(binomial(n-1, 2)/(6*n)) (in lowest terms), n >= 1.
a(n) = numerator(r(n)), with r(n) = s(1,n) = Sum_{r=1..(n-1)} (r/n)*(r/n - floor(r/n)- 1/2), n >= 1. For other forms see the above comments.
PROG
(Julia)
using Nemo
A264388(n) = numerator(dedekind_sum(1, n))
[A264388(n) for n in 1:70] |> println # Peter Luschny, Mar 13 2018
CROSSREFS
KEYWORD
nonn,easy,frac
AUTHOR
Wolfdieter Lang, Jan 11 2016
STATUS
approved