[seqfan] Re: double factorial division

[Luca Petrone]

Dear All,

the sequence of fractions is oscillating and approximating to Sqrt[2/Pi*n] and Sqrt[Pi/2*n] for n odd and n even, respectively: I suppose this is a known result, isn't it?

Best Regards,
Luca Petrone
> Il 25/11/2020 01:46 Neil Sloane <njasloane at gmail.com> ha scritto:
> 
>  
> Anchar, I think there should be three sequences, depending on whether you
> take the floor, the ceiling, or the nearest integer.
> 
> 
> So please go ahead and submit them.
> 
> Here is what I get with Maple:
> 
> > s1:=[seq(f(n),n=1..20)];
> 
> [1, 2, 3/2, 8/3, 15/8, 16/5, 35/16, 128/35, 315/128, 256/63, 693/256,
> 1024/231
> , 3003/1024, 2048/429, 6435/2048, 32768/6435, 109395/32768, 65536/12155,
> 230945/65536, 262144/46189]
> 
> > s2:=evalf(s1);
> s2 := [1., 2., 1.500000000, 2.666666667, 1.875000000, 3.200000000,
> 2.187500000,
> 
>     3.657142857, 2.460937500, 4.063492063, 2.707031250, 4.432900433,
> 
>     2.932617188, 4.773892774, 3.142089844, 5.092152292, 3.338470459,
> 
>     5.391690662, 3.523941040, 5.675463855]
> 
> > s3:=map(floor,s2);
>       s3 := [1, 2, 1, 2, 1, 3, 2, 3, 2, 4, 2, 4, 2, 4, 3, 5, 3, 5, 3, 5]
> 
> > s4:=map(ceil,s2);
>       s4 := [1, 2, 2, 3, 2, 4, 3, 4, 3, 5, 3, 5, 3, 5, 4, 6, 4, 6, 4, 6]
> 
> > s5:=map(round,s2);
>       s5 := [1, 2, 2, 3, 2, 3, 2, 4, 2, 4, 3, 4, 3, 5, 3, 5, 3, 5, 4, 6]
> 
> 
> All three of s2, s3, s4 seem to be new.
> 
> But the numerators and denominators separately  are in the OEIS already,
> see A004730 and A004731 (based on the reciprocals of your fractions, but
> close enough)
> 
> 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, Nov 24, 2020 at 6:37 PM Anchar Koops <anchar.koops at gmail.com> wrote:
> 
> > Dear all,
> >
> >
> >
> > I found an interesting sequence that is not in the OEIS. It is like
> > factorial but with division instead e.g. input 4 would give 4/ (3/ (2/1)) =
> > 2,666…
> >
> > Or:
> >
> > output = n / ((n-1) / ((n-2) / ((n-3) / ... / (3/ (2/1))…)))
> >
> >
> > I was happy to find it can also be written as:
> >
> > output = n!! / (n-1)!!
> >
> >
> >
> > These are the first 10 possible positive integer inputs:
> >
> > n             output                  |output|
> >
> > 1             1,00                      1
> >
> > 2             2,00                      2
> >
> > 3             1,50                      2
> >
> > 4             2,67                      3
> >
> > 5             1,88                      2
> >
> > 6             3,20                      3
> >
> > 7             2,19                      2
> >
> > 8             3,66                      4
> >
> > 9             2,46                      2
> >
> > 10           4,06                      4
> >
> >
> >
> > As you can see the output oscillates. However, some start inputs e.g.
> > 1,3992863… create a line.
> >
> >
> >
> > I cannot imagine this has not been found many times before. Is this
> > sequence worthy?
> >
> >
> >
> > Thank you very much for your attention.
> > Anchar
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
> 
> --
> Seqfan Mailing list - http://list.seqfan.eu/