A128839 Numerators of the continued fraction convergents of the decimal concatenation of the odd natural numbers.

1, 2, 3, 8, 11, 140, 151, 442, 1477, 1919, 11072, 12991, 24063, 37054, 98171, 429738, 957647, 6175620, 13308887

Offset

1,2

Comments

The 19th convergent breaks down at number 95 so a 16 digit ratio gives 91 digits accuracy!

Links

Table of n, a(n) for n=1..19.

Formula

The odd natural numbers 1,3,5.. are concatenated and then preceded by a decimal point to create the fraction N = .1357911131517... . This number is then evaluated with n=1,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 19th convergent 13308887/98010000 =
0.13579111315171921232527293133353739414345474951535557596163656769717375777981\
8385878991939598000...

Prog

(PARI) catodd(n) = f="."; forstep(x=1, 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(numer", "); ) }

Crossrefs

Sequence in context: A139447 A042783 A042583 * A368357 A192113 A243192

Adjacent sequences: A128836 A128837 A128838 * A128840 A128841 A128842

Keyword

frac,nonn,base

Author

Cino Hilliard, Apr 16 2007

Extensions

Edited by Charles R Greathouse IV, Apr 25 2010

Status

approved