[seqfan] Re: OEIS mentioned in What's Special About This Fraction?

Alonso Del Arte alonso.delarte at gmail.com
Tue Dec 8 03:28:33 CET 2020


> (these numerators and denominators are certainly in OEIS... but it would
be
nice to have a direct link for the "frac" sequences to be displayed as
fractions.

I agree 100%. But I know next to nothing about the OEIS behind the scenes
to say how feasible this was back then or is now.

Al

On Mon, Dec 7, 2020 at 4:00 PM M. F. Hasler <oeis at hasler.fr> wrote:

> Pierre wrote:
>
> >   On Friday, December 4, 2020 3:53:45 PM EST Alonso Del Arte wrote:
> > > I've started What's Special About This Fraction?
> > > https://alonso-del-arte.github.io/misc-info/math/fractions.html
> > > The page is a little bare right now, ... There are probably other OEIS
> > > entries that I should mention in that page.
> > A few more rational approximations:
> > 71/41 is an approximation to √3
> > 41/29 is an approximation to √2 (and produces diagonal stripes in Halton,
>
>
> Anyway we have the rational approximations of irrationals from the
> continued fraction expansions:
> (PARI) cf(x)=[c[1]/c[2] | c <- contfracpnqn(contfrac(x),9)]
>
> cf(sqrt(2)) :
> [1/1 ; 3 / 2 ; 7 / 5 ; 17 / 12 ; 41 / 29 ; 99 / 70 ; 239 / 169 ; 577 / 408
> 1393 / 985 ; 3363 / 2378 ]
> cf(sqrt(3))
>  [1, 2, 5/3, 7/4, 19/11, 26/15, 71/41, 97/56, 265/153, 362/209]
>  cf(sqrt(5))
>  [2, 9/4, 38/17, 161/72, 682/305, 2889/1292, 12238/5473, 51841/23184,
> 219602/98209, 930249/416020]
>  cf(Pi)
>  [3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317,
> 312689/99532, 833719/265381, 1146408/364913]
>  cf(exp(1))
>  [2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536]
>  cf(Euler) \\ i.e., gamma
>  [0, 1, 1/2, 3/5, 4/7, 11/19, 15/26, 71/123, 228/395, 3035/5258]
>  cf((sqrt(5)-1)/2) \\ golden ratio phi = 1/Phi = Phi - 1
>  [0, 1, 1/2, 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 34/55]
>  cf(sqrt(Pi))
>  [1, 2, 7/4, 16/9, 23/13, 39/22, 257/145, 296/167, 8545/4821, 111381/62840]
>  cf(sqrt(exp(1)))
>  [1, 2, 3/2, 5/3, 28/17, 33/20, 61/37, 582/353, 643/390, 1225/743]
>  cf(sqrt(Euler))
>  [0, 1, 3/4, 19/25, 117/154, 604/795, 721/949, 217625/286444,
> 435971/573837, 10680929/14058532]
> etc.
> (these numerators and denominators are certainly in OEIS... but it would be
> nice to have a direct link for the "frac" sequences to be displayed as
> fractions. I suggested a mechanism for that several years ago, see
>
> https://oeis.org/wiki/User:M._F._Hasler/Work_in_progress/Improvements_of_OEIS#I._Keywords_with_parameters
> .)
>
> Is it remarkable that the fractions of Pi grow the fastest (among these
> "random samples")...?
> For the golden ratio Phi it is not surprising the fractions are the
> smallest, it's known that this is the "most irrational number" with
> contfrac 1+1/(1+1/(1+1/(1+...))).
>
> - Maximilian
>
>  therefore 71/29 is an approximation to √6
> > 99/70 is an approximation to √2
> >
> > Pierre
> > --
> > La sal en el mar es más que en la sangre.
> > Le sel dans la mer est plus que dans le sang.
> >
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


-- 
Alonso del Arte
Author at SmashWords.com
<https://www.smashwords.com/profile/view/AlonsoDelarte>
Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>



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