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%I #4 Oct 01 2013 21:35:21
%S 40,41,81,1175,4781,10737,26255,36992,840079,877071,15750286,16627357,
%T 49005000
%N Denominators of the continued fraction convergents of the decimal concatenation of the even natural numbers.
%C The 15 digit ratio of the 13th convergent gives an accuracy of 93 digits in the expansion.
%F The even natural numbers 0,2,4.. are concatenated and then preceded by a decimal point to create the fraction N = .024681012... . This number is then evaluated with n=0,m=steps to iterate,x = N, a(0)=floor(N) using the loop: do a(n)=floor(x) x=1/(x-a(n)) n=n+1 loop until n=m
%e The 13th convergent 1209493/49005000 =
%e 0.02468101214161820222426283032343638404244464850525456586062646668707274767880\
%e 8284868890929496990...
%o (PARI) cateven(n) = f=".";forstep(x=0,n,2,a=concat(f,Str(x)));f=eval(f) cfrac2(m,f) = { default(realprecision,1000); cf = vector(m+10); cf = contfrac(f); for(m1=1,m-1, r=cf[m1+1]; forstep(n=m1,1,-1, r = 1/r; r+=cf[n];); numer=numerator(r); denom=denominator(r); print1(denom","); ) }
%K frac,nonn,base
%O 0,1
%A _Cino Hilliard_, Apr 16 2007
%E Edited by _Charles R Greathouse IV_, Apr 25 2010
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