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

0, 1, 1490845, 2981691, 16399300, 35780291, 52179591, 609755792, 661935383, 1271691175, 1933626558, 10939823965, 12873450523, 23813274488, 36686725011, 60499999499

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