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 A101629 #20 Dec 01 2024 20:09:46 %S A101629 1,47,6931,238657,4563655,526760263,45934377581,2852342564497, %T A101629 105651280880749,4335127472172929,186521117762900387, %U A101629 61393482232562091673,3255023127143379846869,3255958701070954680689 %N A101629 Numerator of partial sums of a certain series. %C A101629 The denominators are given in A101630. %C A101629 Third member (m=4) of a family defined in A101028. %C A101629 The limit s=lim(s(n),n->infinity) with the s(n) defined below equals 15*sum(Zeta(2*k+1)/4^(2*k),k=1..infinity) with Euler's (or Riemann's) zeta function. This limit is 15*(3*log(2)-2) = 1.1916231251...; see the Abramowitz-Stegun reference (given in A101028) p. 259, eq. 6.3.15 with z=1/4 together with p. 258. %H A101629 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. %H A101629 W. Lang: <a href="/A101629/a101629.txt">Rationals s(n) and more.</a> %F A101629 a(n)=numerator(s(n)) with s(n)=60*sum(1/((4*k-1)*(4*k)*(4*k+1)), k=1..n) = 15*sum(1/((4*k-1)*k*(4*k+1)), k=1..n). %e A101629 s(3)= 60*(1/(3*4*5)+ 1/(7*8*9) + 1/(11*12*13)) = 6931/6006, hence %e A101629 a(3)=6931 and A101630(3)=6006. %Y A101629 Cf. A101028, A101627, A101631, members 2, 3, 5, resp. %K A101629 nonn,frac,easy %O A101629 1,2 %A A101629 _Wolfdieter Lang_, Dec 23 2004