%I A121000
%S A121000 1,325,52651,34117853,5527092193,596925956851,96702005009873,
%T A121000 125325798492795551,60908338067498638501,19734301533869558876755,
%U A121000 3196956848486868538038509,2071628037819490812648983225
%N A121000 Numerators of partial sums of Catalan numbers scaled by powers of 1/18^2 
               = 1/324.
%C A121000 Denominators are given under A121001.
%C A121000 This is the fourth member (p=3) of the first p-family of partial sums 
               of normalized scaled Catalan series CsnI(p):=sum(C(k)/L(2*p)^(2*k),
               k=0..infinity) with limit L(2*p)*(F(2*p+1) - F(2*p)*phi) = L(2*p)/
               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 A121000 The partial sums of the above mentioned first p-family are rI(p;n):=sum(C(k)/
               L(2*p)^(2*k),k=0..n), n>=0, for p=0,1,...
%C A121000 For more details on this p-family and the other three ones see the W. 
               Lang link under A120996.
%C A121000 The limit lim_{n->infinity} r(n) = 18*(13 - 8* phi) = 18/phi^6 = 1.003105620014 
               (maple10, 15 digits).
%H A121000 W. Lang: <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A121000.text">
               Rationals r(n), limit.</a>
%F A121000 a(n)=numerator(r(n)) with r(n) := rI(p=3,n) = sum(C(k)/L(6)^(2*k),k=0..n), 
               with Lucas L(6)=18 and C(k):=A000108(k) (Catalan). The rationals 
               r(n) are given in lowest terms.
%e A121000 Rationals r(n): [1, 325/324, 52651/52488, 34117853/34012224,
%e A121000 5527092193/5509980288, 596925956851/595077871104, ...].
%Y A121000 Sequence in context: A031606 A145414 A166220 this_sequence A048909 A097739 
               A048918
%Y A121000 Adjacent sequences: A120997 A120998 A120999 this_sequence A121001 A121002 
               A121003
%K A121000 nonn,frac,easy
%O A121000 0,2
%A A121000 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 16 
               2006