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 A121010 #8 Aug 30 2019 03:49:09
%S A121010 1,319,51041,6533247,5226597607,1672511234219,267601797475073,
%T A121010 342530300768093011,2192193924915795299,17537551399326362389569,
%U A121010 2806008223892217982335239,1795845263291019508694523567
%N A121010 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(5*8^2) = 1/320.
%C A121010 Denominators are given under A121011.
%C A121010 This is the third member (p=3) of the third p-family of partial sums of normalized scaled Catalan series CsnIII(p):=sum(((-1)^k)*C(k)/((5^k)*F(2*p)^(2*k)),k=0..infinity) with limit F(2*p)*(-L(2*p+1) + L(2*p)*phi) = F(2*p)*sqrt(5)/phi^(2*p), with C(n)=A000108(n) (Catalan), F(n)= A000045(n) (Fibonacci), L(n) = A000032(n) (Lucas) and phi:=(1+sqrt(5))/2 (golden section).
%C A121010 The partial sums of the above mentioned third p-family are rIII(p;n):=sum(((-1)^k)*C(k)/((5^k)*F(2*p)^(2*k)),k=0..n), n>=0, for p=1,...
%C A121010 For more details on this p-family and the other three ones see the W. Lang link under A120996.
%H A121010 W. Lang: <a href="/A121010/a121010.txt">Rationals r(n), limit.</a>
%F A121010 a(n)=numerator(r(n)) with r(n) := rIII(p=3,n) = sum(((-1)^k)*C(k)/((5^k)*F(2*3)^(2*k)),k=0..n), with F(6)=8 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.
%e A121010 Rationals r(n): [1, 319/320, 51041/51200, 6533247/6553600,
%e A121010 5226597607/5242880000, 1672511234219/1677721600000,...].
%p A121010 The limit lim_{n->infinity} (r(n) := rIII(3;n)) = 8*(-29 + 18*phi) = 8*sqrt(5)/phi^6 = 0.9968943824 (maple10, 10 digits).
%Y A121010 The second member is A121008/A121009.
%K A121010 nonn,frac,easy
%O A121010 0,2
%A A121010 _Wolfdieter Lang_, Aug 16 2006