%I A120996
%S A120996 1,10,92,833,7511,22547,202967,1826846,49326272,443941310,3995488586,
%T A120996 35959456060,323635312552,2912718555868,2912718853028,26214470754457,
%U A120996 235930240718743,6370116542620991,57331049042801819
%N A120996 Numerators of partial sums of Catalan numbers scaled by powers of 1/9.
%C A120996 Denominators are given under A120997.
%C A120996 This is the second member (p=1) 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 A120996 Other members of this rational p-family are: A120778(n)/A120777(n) (p=0), 
               ......
%C A120996 From the expansion of sqrt(1-x) = 1 -(x/2)*sum(C(k)*(x/4)^k,k=0..infinity), 
               for |x|<=1, one has sum(C(k)/q^k,k=0..infinity) = (q - sqrt(q*(q-4)))/
               2, for |q|>=4.
%C A120996 The CsnI(1) series value is lim_{n->infinity}(r(n)) = 3*(2-phi) = 3/phi^2 
               = 1.145898034 (maple10, 10 digits).
%C A120996 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 A120996 For special q values q(n):=2 + L(2*n), n>=0, one finds the above given 
               limits for the p-family members for n=2*p (even), p>=0. The Pell 
               equation x^2 -5*y^2 = +4 with general (nonnegative) solutions (x,
               y)=(L(2*n),F(2*n)) and well known identities for Fibonacci and Lucas 
               numbers were used to derive the above given p-family result. See 
               the W. Lang link.
%H A120996 W. Lang: <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A120996.text">
               Rationals r(n), limit and four p-families of scaled Catalan series.</
               a>
%F A120996 a(n)=numerator(r(n)) with r(n) := rI(p=1,n) = sum(C(k)/L(2)^(2*k),k=0..n), 
               with Lucas L(2)=3 and C(k):=A000108(k) (Catalan). The rationals r(n) 
               are given in lowest terms.
%e A120996 Rationals r(n): [1, 10/9, 92/81, 833/729, 7511/6561, 22547/19683,
%e A120996 202967/177147, 1826846/1594323,...].
%Y A120996 Sequence in context: A052266 A027325 A037534 this_sequence A103944 A099295 
               A167589
%Y A120996 Adjacent sequences: A120993 A120994 A120995 this_sequence A120997 A120998 
               A120999
%K A120996 nonn,frac,easy
%O A120996 0,2
%A A120996 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 16 
               2006