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 A101631 #22 Dec 01 2024 20:15:02
%S A101631 1,37,1069,20575,1346153,1214756107,20699705479,850029466379,
%T A101631 19572345658457,137116980686111,411600123273343,1482039573988769177,
%U A101631 456179332236626381,32398234503565880731,1199020509231104363863
%N A101631 Numerator of partial sums of a certain series.
%C A101631 The denominators are given in A101632.
%C A101631 Third member (m=5) of a family defined in A101028.
%C A101631 The limit s = lim_{n->oo} s(n) with the s(n) defined below equals 24*Sum_{k>=1} zeta(2*k+1)/5^(2*k) with Euler's (or Riemann's) zeta function. This limit is -24*(gamma + Psi(1/5) + 5/2 + Pi*cot(Pi/5)/2) = 1.1954056019...; see a comment in A101028 following from the Abramowitz-Stegun reference (given in A101028) p. 259, eq. 6.3.15 with z=1/5 together with p. 258.
%H A101631 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 A101631 W. Lang: <a href="/A101631/a101631.txt">Rationals s(n,5) and more.</a>
%F A101631 a(n) = numerator(s(n)) where s(n) = 120*Sum_{k=1..n} 1/((5*k-1)*(5*k)*(5*k+1)) = 24*Sum_{k=1..n} 1/((5*k-1)*k*(5*k+1)).
%e A101631 s(3) = 120*(1/(4*5*6) + 1/(9*10*11) + 1/(14*15*16)) = 1069/924, hence a(3)=1069 and A101632(3)=924.
%Y A101631 Cf. A101028, A101627, A101629, members 2, 3, 4, resp.
%K A101631 nonn,frac,easy
%O A101631 1,2
%A A101631 _Wolfdieter Lang_, Dec 23 2004