[seqfan] Re: double factorial division

[Neil Sloane]

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
>
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>