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

1,5

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.

Formula

a(n) = numerator((n-1)*(n-3)/(6*(2*n-1)) (in lowest terms), 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

Mathematica

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

Prog

(PARI) a(n)=numerator((n-1)*(n-3)/(12*n-6)) \\ Charles R Greathouse IV, Nov 28 2016
(Magma) [Numerator((n-1)*(n-3)/(6*(2*n-1))): n in [1..60]]; // Vincenzo Librandi, Nov 21 2018

Crossrefs

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

Sequence in context: A021877 A355677 A359179 * A200623 A248671 A343442

Adjacent sequences: A278710 A278711 A278712 * A278714 A278715 A278716

Keyword

sign,frac,easy

Author

Wolfdieter Lang, Nov 28 2016

Status

approved