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A128837 Numerator of the continued fraction convergents of the decimal concatenation of the natural numbers. 0

%I #9 Jan 06 2022 08:14:48

%S 0,1,1490845,2981691,16399300,35780291,52179591,609755792,661935383,

%T 1271691175,1933626558,10939823965,12873450523,23813274488,

%U 36686725011,60499999499

%N Numerator of the continued fraction convergents of the decimal concatenation of the natural numbers.

%C The 15th convergent breaks down at number 16 so a 24-digit ratio gives 24 digits accuracy. The 16th convergent breaks down at the 97th number. It is amazing that a 24-digit ratio gives 186 digits of accuracy in the expansion!

%F The natural numbers 0,1,2,3,... are concatenated and then preceded by a decimal point to create the fraction N = .0123456789101112131415... . 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 15th convergent 36686725011/2971624750499 = 0.01234567891011121314151610314942472616...

%e The 16th convergent 60499999499/4900500000000 = 0.0123456789101112131415161718192021222324252627282930313233343536373839404142\

%e 434445464748495051525354555657585960616263646566676869707172737475767778798081\

%e 8283848586878889909192939495969799000...

%o (PARI) cfrac2(m,f) = { default(realprecision,1000); cf = vector(m+10); cf = contfrac(f); for(m1=0,m-1, r=cf[m1+1]; forstep(n=m1,1,-1, r = 1/r; r+=cf[n];); numer=numerator(r); denom=denominator(r); print1(numer","); ) }

%K frac,nonn,base

%O 0,3

%A _Cino Hilliard_, Apr 15 2007

%E Edited by _Charles R Greathouse IV_, Apr 25 2010

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Last modified August 31 10:25 EDT 2024. Contains 375560 sequences. (Running on oeis4.)