0,2

Denominators are given under A120997.

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).

Other members of this rational p-family are: A120778(n)/A120777(n) (p=0), ......

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.

The CsnI(1) series value is lim_{n->infinity}(r(n)) = 3*(2-phi) = 3/phi^2 = 1.145898034 (maple10, 10 digits).

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,...

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.

W. Lang: Rationals r(n), limit and four p-families of scaled Catalan series.

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.

Rationals r(n): [1, 10/9, 92/81, 833/729, 7511/6561, 22547/19683,

202967/177147, 1826846/1594323,...].

Sequence in context: A052266 A027325 A037534 this_sequence A103944 A099295 A167589

Adjacent sequences: A120993 A120994 A120995 this_sequence A120997 A120998 A120999

nonn,frac,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 16 2006