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W. Wolfdieter Lang: , <a href="/A120794/a120794.txt">Rationals r(n) and limit.</a>
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From the expansion of sqrt(1+1/4) = 1+(1/8)*sum(Sum_{k>=0} C(k)/(-16)^k,k=0..infinity) one has, with the partial sums r(n) are defined below, r :=limit( lim_{n->oo} r(n),n to infinity) = 4*(sqrt(5)-2) = 4*(2*phi-3)) = 0.944271909...
This is the first member (p=1) of the fourth famliy family of scaled Catalan sums with limits in Q(sqrt(5)). See the W. Lang link under A120996.
a(n)=numerator(r(n)), with the rationals r(n):=sum(Sum_{k=0..n} ((-1)^k)*C(k)/16^k,k=0..n) with C(k):=A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.
Rationals r(n): [1, 15/16, 121/128, 3867/4096, 30943/32768, 495067/524288, 3960569/4194304,...].
495067/524288, 3960569/4194304,...].
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W. Lang: <a href="/LANGCHANGE/A120794/a120794.texttxt">Rationals r(n) and limit.</a>
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W. Lang: <a href="http://www.itp.kit.edu/~wl/EISpubLANGCHANGE