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Numerators of partial sums of Catalan numbers scaled by powers of 1/8.
2

%I #6 Aug 30 2019 03:41:51

%S 1,9,37,597,2395,19181,76757,2456653,9827327,78621047,314488387,

%T 5031843585,20127426343,161019596469,644078720181,41221047786429,

%U 164884208824551,1319073735418803,5276295061084887,84420721860989787

%N Numerators of partial sums of Catalan numbers scaled by powers of 1/8.

%C Denominators are under A120781.

%C 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 W. Lang: <a href="/A120780/a120780.txt">Rationals r(n) and limit.</a>

%F 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 Rationals r(n): [1, 9/8, 37/32, 597/512, 2395/2048, 19181/16384,

%e 76757/65536, 2456653/2097152,...].

%K nonn,easy,frac

%O 0,2

%A _Wolfdieter Lang_, Jul 20 2006