This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A278713 #29 Feb 16 2025 08:33:37
%S A278713 0,-1,0,1,4,5,4,7,8,21,40,33,4,143,28,65,112,17,48,323,60,133,44,161,
%T A278713 88,575,104,45,364,261,140,899,32,341,544,385,204,259,228,481,760,533,
%U A278713 56,1763,308,645,1012,141,368,2303,400,833,260,901,468,2915,504,209
%N A278713 Numerators of (n-1)*(n-3)/(6*(2*n-1)); equivalently, numerators of Dedekind sum s(2,2*n-1).
%C A278713 For the denominators see A278714.
%C A278713 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.
%D A278713 Apostol, Tom, M., Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990.
%D A278713 Ayoub, R., An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963, pp. 168, 191.
%H A278713 Vincenzo Librandi, <a href="/A278713/b278713.txt">Table of n, a(n) for n = 1..10000</a>
%H A278713 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DedekindSum.html">Dedekind Sum</a>.
%F A278713 a(n) = numerator((n-1)*(n-3)/(6*(2*n-1))).
%F A278713 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.
%F A278713 (n-1)*(n-3)/30 <= a(n) <= (n-1)*(n-3) for n > 2. - _Charles R Greathouse IV_, Nov 28 2016
%t A278713 Table[Numerator[(n - 1) (n - 3) / (6 (2 n - 1))], {n, 60}] (* _Vincenzo Librandi_, Nov 21 2018 *)
%o A278713 (PARI) a(n)=numerator((n-1)*(n-3)/(12*n-6)) \\ _Charles R Greathouse IV_, Nov 28 2016
%o A278713 (Magma) [Numerator((n-1)*(n-3)/(6*(2*n-1))): n in [1..60]]; // _Vincenzo Librandi_, Nov 21 2018
%Y A278713 Cf. A278714, A264388/A264389 for s(1,n).
%K A278713 sign,frac,easy
%O A278713 1,5
%A A278713 _Wolfdieter Lang_, Nov 28 2016