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Search: id:A128837
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| A128837 |
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Numerator of the continued fraction convergents of the decimal concatenation of the natural numbers. |
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+0
1
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| 0, 1, 1490845, 2981691, 16399300, 35780291, 52179591, 609755792, 661935383, 1271691175, 1933626558, 10939823965, 12873450523, 23813274488, 36686725011, 60499999499
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OFFSET
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0,3
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COMMENT
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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!
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FORMULA
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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
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EXAMPLE
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The 15th convergent
36686725011/2971624750499 = 0.01234567891011121314151610314942472616...
The 16th convergent 60499999499/4900500000000 =
0.0123456789101112131415161718192021222324252627282930313233343536373839404142\
434445464748495051525354555657585960616263646566676869707172737475767778798081\
8283848586878889909192939495969799000...
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PROGRAM
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(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", "); ) }
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CROSSREFS
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Sequence in context: A003932 A159345 A126175 this_sequence A126252 A089492 A151620
Adjacent sequences: A128834 A128835 A128836 this_sequence A128838 A128839 A128840
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KEYWORD
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frac,nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)hotmail.com), Apr 15 2007
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