Search: id:A123747
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%I A123747
%S A123747 1,7,41,9,239,6227,32059,163727,166301,841229,21215481,106782837,
%T A123747 536618341,538698461,172897,13538601629,67813224223,339532842359,
%U A123747 339895847771,1700893049407,42549895540939,212857129279583
%N A123747 Numerators of partial sums of a series for sqrt(5).
%C A123747 Denominators are given by A123748.
%C A123747 The sum over central binomial coefficients scaled by powers of 5, r(n):=sum(binomial(2*k,
               k)/5^k,k=0..n) has the limit s:=lim(r(n),n->infinity) = sqrt(5). 
               From the expansion of 1/sqrt(1-x) for x=4/5.
%H A123747 W. Lang: <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A123747.text">
               Rationals and more.</a>
%F A123747 a(n)=numerator(r(n)) with the rationals r(n):=sum(binomial(2*k,k)/5^k,
               k=0..n) in lowest terms.
%F A123747 r(n)=sum(((2*k-1)!!/((2*k)!!)*(4/5)^k,k=0..n),n>=0, with the double factorials 
               A001147 and A000165.
%e A123747 a(3)=9 because r(3)= 1+2/5+6/25+4/25 = 9/5 = a(3)/A123748(3).
%Y A123747 Cf. A001077/A001076 continued fraction convergents for sqrt(5).
%Y A123747 Sequence in context: A121582 A062727 A165397 this_sequence A144421 A023251 
               A073501
%Y A123747 Adjacent sequences: A123744 A123745 A123746 this_sequence A123748 A123749 
               A123750
%K A123747 nonn,frac,easy
%O A123747 0,2
%A A123747 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Nov 10 2006

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