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A278713
Numerators of (n-1)*(n-3)/(6*(2*n-1)); equivalently, numerators of Dedekind sum s(2,2*n-1).
3
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
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
KEYWORD
sign,frac,easy
AUTHOR
Wolfdieter Lang, Nov 28 2016
STATUS
approved