[seqfan] Re: double factorial division

jean-paul allouche jean-paul.allouche at imj-prg.fr
Wed Nov 25 18:56:13 CET 2020

```Hi

Writing the division(s) as a single division of the form (x.y.z.../a.b.c...)
one can see the relation with Wallis formula (hence the Srqt(2/Pi*n) and
Sqrt(Pi/2*n))
best
jean-paul

Le 25/11/2020 à 13:15, Luca Petrone via SeqFan a écrit :
> 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.
>>
>>
>>
>> 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.
>> 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/
>>>
>> --
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