A278713 Numerators of (n-1)*(n-3)/(6*(2*n-1)); equivalently, numerators of Dedekind sum s(2,2*n-1).

0, -1, 0, 1, 4, 5, 4, 7, 8, 21, 40, 33, 4, 143, 28, 65, 112, 17, 48, 323, 60, 133, 44, 161, 88, 575, 104, 45, 364, 261, 140, 899, 32, 341, 544, 385, 204, 259, 228, 481, 760, 533, 56, 1763, 308, 645, 1012, 141, 368, 2303, 400, 833, 260, 901, 468, 2915, 504, 209

Offset: 1

Author

Wolfdieter Lang, Nov 28 2016

Keywords

• sign
• frac
• easy

Comments

For the denominators see A278714.

This gives the numerators of the rational numbers r(n) = s(2,2*n-1), 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 references, Apostol pp. 52, 61-69, 72-73, Ayoub, p. 168, and the Weisstein link. Because gcd(h,k) = 1 is assumed, for h=2 only odd k is of interest.

References

• Apostol, Tom, M., Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990.
• Ayoub, R., An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963, pp. 168, 191.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Dedekind Sum.

Crossrefs

Cf. A278714, A264388/A264389 for s(1,n).

Programs

Magma

[Numerator((n-1)*(n-3)/(6*(2*n-1))): n in [1..60]]; // Vincenzo Librandi, Nov 21 2018

Mathematica

Table[Numerator[(n - 1) (n - 3) / (6 (2 n - 1))], {n, 60}] (* Vincenzo Librandi, Nov 21 2018 *)

PARI

a(n)=numerator((n-1)*(n-3)/(12*n-6)) \\ Charles R Greathouse IV, Nov 28 2016

Formula

a(n) = numerator((n-1)*(n-3)/(6*(2*n-1))).

a(n) = numerator(r(n)), with r(n) = s(2,2*n-1) where s(2,k) = Sum_{r=1..(k-1)} (r/k)*(2*r/k - floor(2*r/k)- 1/2), for odd k.

(n-1)*(n-3)/30 <= a(n) <= (n-1)*(n-3) for n > 2. - Charles R Greathouse IV, Nov 28 2016