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

Neil Sloane njasloane at gmail.com
Tue Dec 8 20:10:26 CET 2020


I should have been more explicit.
The index to fractions is the file
https://oeis.org/wiki/Index_to_fractions_in_OEIS

and here is how it starts:

(after the introduction)

List of fractions[edit
<https://oeis.org/w/index.php?title=Index_to_fractions_in_OEIS&action=edit&section=3>
]First nontrivial term <= 1/3[edit
<https://oeis.org/w/index.php?title=Index_to_fractions_in_OEIS&action=edit&section=4>
]

   - 1/192, 1/21504, 1/190080, 27/573440, 127/218880, 34603/40480,
   278617/2106, 264156210586399/53329920, ... = A048615
   <http://oeis.org/A048615>/A048616 <http://oeis.org/A048616> (Hadamard
   mass)
   - 1, 1/48, 77/7680, 17017/1105920, 52055003/1061683200,
   1509595087/5662310400, 3603403472669/1630745395200,
   10151817126357907/391378894848000, ... = A226256
   <http://oeis.org/A226256>/A226257 <http://oeis.org/A226257>
   - 1, 1/12, 1/288, -139/51840, -571/2488320, 163879/209018880,
   5246819/75246796800, -534703531/902961561600, ... = A001163
   <http://oeis.org/A001163>/A001164 <http://oeis.org/A001164> (Stirling
   approximation)
   - 1, -1/12, 7/240, -31/1344, 127/3840, -2555/33792, 1414477/5591040,
   -57337/49152, 118518239/16711680, ... = A001896 <http://oeis.org/A001896>
   /A033469 <http://oeis.org/A033469> (Bernoulli(2n,1/2))
   - 1/10, 3/10, 567/130, 43659/170, 392931/10, ... = A002306
   <http://oeis.org/A002306>/A047817 <http://oeis.org/A047817> (Hurwitz
   numbers H_n))
   - 1/9, 10/81, 100/729, 1000/6561, 10000/59049, ... = A100061
   <http://oeis.org/A100061>/A100062 <http://oeis.org/A100062>
   - 1/8, 1/12, 11/96, 17/72, 619/960, 709/324, ... = A226258
   <http://oeis.org/A226258>/A226259 <http://oeis.org/A226259>
   - 1, 1/8, 35/384, 385/3072, 25025/98304, 1616615/2359296,
   260275015/113246208, 929553625/100663296, 835668708875/19327352832, ... =
   A225697 <http://oeis.org/A225697>/A225698 <http://oeis.org/A225698>
   - 1/8, 16/45, 25/144, 34/105, 2989/17280, 5248/14175, 1209/5600,
   5675/12474, 560593/1935360, 893128/1576575, 11148172711/28740096000,
   109420087/156370500, ... = A100647 <http://oeis.org/A100647>/A100648
   <http://oeis.org/A100648> (Cotesian C(n,3))
   - 1, -1/7, 1/49, -1/343, 1/2401, -1/16807, 1/117649, -1/823543,
   1/5764801, -1/40353607, 1/282475249, -1/1977326743, ... = A033999
   <http://oeis.org/A033999>/A000420 <http://oeis.org/A000420> ( terms of
   the series (-1)^n/7^n, where n=0..inf, which converges to 7/8; actually you
   could expand this with a denominator sequence related to powers of x (
   A001018 <http://oeis.org/A001018>, A001019 <http://oeis.org/A001019>,
   A011557 <http://oeis.org/A011557>, ...) that when applied to the
   infinite series (-1)^n/x^n it will converge to x/(x+1) )
   - 0, 1, 0, -1/6, 0, 1/120, 0, -1/5040, 0, 1/362880, 0, -1/39916800, 0,
   1/6227020800, 0, -1/1307674368000, ... = A033999
   <http://oeis.org/A033999>/A009445 <http://oeis.org/A009445> (sin(x))
   - 1, 0, 1/6, 0, 7/360, 0, 31/15120, 0, 127/604800, 0, 73/3421440, 0,
   1414477/653837184000, 0, 8191/37362124800, 0, 16931177/762187345920000, ...
   = A036280 <http://oeis.org/A036280>/A036281 <http://oeis.org/A036281> (x/sin
   x)
   - 0, 1, 0, 1/6, 0, 3/40, 0, 5/112, 0, 35/1152, 0, 63/2816, 0, 231/13312,
   0, 143/10240, 0, 6435/557056, 0, 12155/1245184, 0, 46189/5505024, 0, ... =
   A055786 <http://oeis.org/A055786>/A002595 <http://oeis.org/A002595>
   - 1, -1/6, 1/120, -1/5040, 1/362880, -1/39916800, 1/6227020800,
   -1/1307674368000, ... = A033999 <http://oeis.org/A033999>/A009445
   <http://oeis.org/A009445> (sin(x))
   - 1/6, 1/90, 1/945, 1/9450, 1/93555, 691/638512875, 2/18243225,
   3617/325641566250, 43867/38979295480125, ... = A046988
   <http://oeis.org/A046988>/A002432 <http://oeis.org/A002432>
   - 1, 1/6, 7/360, 31/15120, 127/604800, 73/3421440, 1414477/653837184000,
   8191/37362124800, ... = A036280 <http://oeis.org/A036280>/A036281
   <http://oeis.org/A036281> (x/sin x)
   - 1, -1/6, 1/36, -1/216, 1/1296, -1/7776, 1/46656, -1/279936, 1/1679616,
   -1/10077696, 1/60466176, -1/362797056, ... = A033999
   <http://oeis.org/A033999>/A000400 <http://oeis.org/A000400> (terms of
   the series (-1)^n/6^n, where n=0..inf, which converges to 6/7)


...

It is a fairly long list.

At the end, there are these instructions:

Comments[edit
<https://oeis.org/w/index.php?title=Index_to_fractions_in_OEIS&action=edit&section=10>
]

   - Updates to this Index are welcomed, but please be very careful.
   Remember that this is a scientific database. Please preserve the ordering
   (which is described at the top of this page).
   - The format is as follows: One line for each sequence of fractions,
   beginning with * followed by a space, then the first few fractions as
   numerator/denominator, separated by commas and spaces, followed by " ... =
   A000001 <http://oeis.org/A000001>/A000002 <http://oeis.org/A000002>"
   (say), where A000001 <http://oeis.org/A000001> and A000002
   <http://oeis.org/A000002> are the A-numbers for the numerator and
   denominator sequences respectively. Unless the definition is very
   complicated, include a brief description in parentheses at the end of the
   line.
   - Several of the lines are too short. If you would like to help, please
   add more terms.
   - Interpolated zeros, and sometimes signs, have been ignored when
   matching with the A-numbers <https://oeis.org/wiki/A-numbers>.
   - When you submit a pair of numerator-and-denominator sequences as new
   sequences to the OEIS, you should include a line like the following in the
   *Example* section of both the numerator and denominator sequences:

1, -1/3, -1/45, -2/945, -1/4725, -2/93555, -1382/638512875, -4/18243225,
-3617/162820783125, -87734/38979295480125, -349222/1531329465290625, ... =
A002431 <http://oeis.org/A002431>/A036278 <http://oeis.org/A036278>and make
a corresponding entry here.

   - Fractions are written as 7/4 (say), rather than 1 3/4 (i.e. no mixed
   fractions <https://oeis.org/wiki/Mixed_fractions>).
   - Fractions are normally in lowest terms
   <https://oeis.org/wiki/Lowest_terms>, with any common factors canceled
   out.
   - All sequences mentioned here should have keyword *frac*
   <https://oeis.org/wiki/Category:Keyword_frac> in the OEIS.
   - Do not sign changes to this index with your name (the history tab will
   show who made the changes).



   - *Please be VERY CAREFUL when making changes!* *If in doubt, consult
   one of the Editors-in-Chief <https://oeis.org/wiki/Editors-in-Chief> before
   making any changes.*


*---------*
end of extracts

This Index - which editors don't seem to know about - needs updating

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 Tue, Dec 8, 2020 at 1:52 PM M. F. Hasler <oeis at hasler.fr> wrote:

> On Mon, 7 Dec 2020, 23:20 Neil Sloane, <njasloane at gmail.com> wrote:
>
> > The Index to Fractions in the OEIS on the OEIS wiki does that:  it should
> > the fractions written as fractions
>
>
> Well, just to be clear, by "displayed as fractions" I didn't mean Axxx/Ayyy
> but
> a(1)/b(1), a(2)/b(2), a(3)/b(3), ... with the actual numbers.
> It would be easy with "extended keywords" (cf. OEIS wiki link on my
> previous mail):
> the keyword "frac:Ayyy" could be displayed as a link or button executing a
> JavaScript (approximately):
>
> var Ayyy=getSection(this,"keywords").innerText.pregReplace(
> /.*frac:(A[0-9]+).*/, '\1' );
> getSection(this,"data").innerHTML += "<br/>as fractions with denominator
> "+Ayyy+": " + makeFrac( getSection(this,"data").innerText,getSeqData( Ayyy
> ));
>
> function getSection(t,n){return t.parentNode.getElementByName(n)}
> function getSeqData( n ){ var s=""; for(var r of getFile("
> https://oeis.org/
> "+n+"/internal"))
>  switch( r.substr(0,2)) case '%S': case '%T': case '%U': s +=
> r.substr(3)+",";
>  return s}
> function makeFrac(a,b) { var s="",A=a.split(','), B=b.split(','); for(var
> i=0; i<min( A.length, B.length); ++i) s+=A[i]+"/"+B[i]+","; return s}
>
> That's all it needs (up to slightly approximative syntax / functions
> pregReplace, getFile and parentNode (due to ill-structured HTML)).
> One could simply print the list of a(n)/b(n) as above, or also print the
> list of b(n)/a(n), or use the 2nd arg to the frac keyword
> (with extended syntax: frac[:Ayyy[:{n|d}]]) to know whether the other
> sequence represents numerators or denominators.
> but since a link to the other seq. is given, just use is as denominators as
> above and the reader can click on the A-number to get the reciprocal
> fractions.)
>
> I will add this ASAP to my https://github.com/m-f-h/OEIS.js which already
> demonstrates similar very easily implemented interactive client-side
> functionality for LREC sequences (automatic extension of DATA and
> generation of PROGRAMs).
>
> - Maximilian
>
> 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
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>



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