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A264388 Numerators of binomial(n-1, 2)/(6*n), for n >= 1. Numerators of Dedekind sum s(1, n). 6
0, 0, 1, 1, 1, 5, 5, 7, 14, 3, 15, 55, 11, 13, 91, 35, 20, 34, 51, 57, 95, 35, 77, 253, 46, 25, 325, 117, 63, 203, 145, 155, 248, 44, 187, 595, 105, 111, 703, 247, 130, 205, 287, 301, 473, 165, 345, 1081, 188, 98, 1225, 425, 221, 689, 477, 495, 770, 133, 551 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..59.

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

Cf. A161680, A264389.

Sequence in context: A252006 A158299 A093307 * A141392 A088047 A109257

Adjacent sequences:  A264385 A264386 A264387 * A264389 A264390 A264391

KEYWORD

nonn,easy,frac

AUTHOR

Wolfdieter Lang, Jan 11 2016

STATUS

approved

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Last modified September 22 05:47 EDT 2019. Contains 327287 sequences. (Running on oeis4.)