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

Neil Sloane njasloane at gmail.com
Tue Dec 8 04:19:46 CET 2020


The Index to Fractions in the OEIS on the OEIS wiki does that:  it should
the fractions written as fractions
Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Mon, Dec 7, 2020 at 9:28 PM Alonso Del Arte <alonso.delarte at gmail.com>
wrote:

> > (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>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>



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