A128842 Numerators of the continued fraction convergents of the decimal concatenation of the even natural numbers.

1, 1, 2, 29, 118, 265, 648, 913, 20734, 21647, 388733, 410380, 1209493

Offset

0,3

Comments

The 15 digit ratio of the 13th convergent gives an accuracy of 93 digits in the expansion.

Links

Table of n, a(n) for n=0..12.

Formula

The even natural numbers 0,2,4.. are concatenated and then preceded by a decimal point to create the fraction N = .024681012... . 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 13th convergent 1209493/49005000 =
0.02468101214161820222426283032343638404244464850525456586062646668707274767880\
8284868890929496990...

Prog

(PARI) cateven(n) = f="."; forstep(x=0, 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: A253305 A123004 A062618 * A308899 A028883 A024200

Adjacent sequences: A128839 A128840 A128841 * A128843 A128844 A128845

Keyword

frac,nonn,base

Author

Cino Hilliard, Apr 16 2007

Extensions

Edited by Charles R Greathouse IV, Apr 25 2010

Status

approved