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

M. F. Hasler oeis at hasler.fr
Mon Dec 7 22:00:04 CET 2020


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.
>
>



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