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 A128837 Numerator of the continued fraction convergents of the decimal concatenation of the natural numbers. 0
 0, 1, 1490845, 2981691, 16399300, 35780291, 52179591, 609755792, 661935383, 1271691175, 1933626558, 10939823965, 12873450523, 23813274488, 36686725011, 60499999499 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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! LINKS Table of n, a(n) for n=0..15. FORMULA 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. EXAMPLE The 15th convergent 36686725011/2971624750499 = 0.01234567891011121314151610314942472616... The 16th convergent 60499999499/4900500000000 = 0.0123456789101112131415161718192021222324252627282930313233343536373839404142\ 434445464748495051525354555657585960616263646566676869707172737475767778798081\ 8283848586878889909192939495969799000... PROG (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", "); ) } CROSSREFS Sequence in context: A234837 A233391 A126175 * A297881 A335289 A228761 Adjacent sequences: A128834 A128835 A128836 * A128838 A128839 A128840 KEYWORD frac,nonn,base AUTHOR Cino Hilliard, Apr 15 2007 EXTENSIONS Edited by Charles R Greathouse IV, Apr 25 2010 STATUS approved

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Last modified July 21 06:08 EDT 2024. Contains 374463 sequences. (Running on oeis4.)