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 A120780 #6 Aug 30 2019 03:41:51 %S A120780 1,9,37,597,2395,19181,76757,2456653,9827327,78621047,314488387, %T A120780 5031843585,20127426343,161019596469,644078720181,41221047786429, %U A120780 164884208824551,1319073735418803,5276295061084887,84420721860989787 %N A120780 Numerators of partial sums of Catalan numbers scaled by powers of 1/8. %C A120780 Denominators are under A120781. %C A120780 From the expansion of sqrt(2)/2 = sqrt(1-1/2) = 1-(1/4)*sum(C(k)/8^k,k=0..infinity) one has r:=limit(r(n),n to infinity)= 2*(2 - sqrt(2)) = 1.171572875..., with the partial sums r(n) defined below. %H A120780 W. Lang: <a href="/A120780/a120780.txt">Rationals r(n) and limit.</a> %F A120780 a(n)=numerator(r(n)), with the rationals r(n):=sum(C(k)/8^k,k=0..n) with C(k):=A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms. %e A120780 Rationals r(n): [1, 9/8, 37/32, 597/512, 2395/2048, 19181/16384, %e A120780 76757/65536, 2456653/2097152,...]. %K A120780 nonn,easy,frac %O A120780 0,2 %A A120780 _Wolfdieter Lang_, Jul 20 2006