|
%I #9 Aug 29 2019 16:25:32
%S 1,4,22,21,539,2653,13397,66556,66842,1666188,8347736,41679894,
%T 208607482,208458902,1042829398,5212208021,26068111639,130314629237,
%U 26066746957,3257989916987,16291262409019
%N Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/5.
%C Denominators are given under A121007.
%C This is the first member (p=1) 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 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 For more details on this p-family and the other three ones see the W. Lang link under A120996.
%H W. Lang: <a href="/A121006/a121006.txt">Rationals r(n), limit.</a>
%F a(n)=numerator(r(n)) with r(n) := rIII(p=1,n) = sum(((-1)^k)*C(k)/5^k,k=0..n) and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.
%e Rationals r(n): 1, 4/5, 22/25, 21/25, 539/625, 2653/3125,
%e 13397/15625, 66556/78125, 66842/78125, 1666188/1953125, 8347736/9765625,...]
%p The limit lim_{n->infinity} (r(n) := rIII(1;n)) = -4 + 3*phi = sqrt(5)/phi^2 = 0.85410196624968 (maple10, 15 digits).
%Y The second member (p=2) is A121008/A121009.
%K nonn,frac,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 16 2006
|
