[seqfan] Re: any value storing constants of power ratios?

[Robert G. Wilson, v]

Dear Richard,

    Here is the Mathematica coding for the sequences below:

f[x_] := RealDigits[ Re[2/x*Product[ Gamma[-(Cos[Pi/x] + 
I*Sin[Pi/x])^j]^(-(-1)^j), {j, 2 x - 1}]], 10, 111][[1]]

    It is  very quick and provides a few more digits than the sequences 
now provide. They are for f(4) to f(10):

{8, 4, 8, 0, 5, 4, 0, 4, 9, 3, 5, 2, 9, 0, 0, 3, 9, 2, 1, 2, 9, 6, 5, 0, 
1, 8, 3, 4, 0, 5, 0, 0, 7, 7, 0, 5, 8, 4, 7, 9, 8, 7, 4, 8, 6, 8, 8, 4, 
7, 1, 7, 6, 6, 6, 4, 3, 0, 6, 9, 6, 4, 5, 3, 8, 0, 6, 6, 1, 3, 5, 7, 2, 
8, 5, 5, 5, 5, 4, 4, 1, 2, 7, 1, 3, 6, 7, 6, 6, 3, 7, 6, 7, 3, 6, 9, 0, 
1, 2, 5, 2, 9, 5, 8, 7, 6, 6, 6, 7, 0, 6, 3}

{9, 2, 8, 7, 8, 6, 9, 3, 5, 7, 9, 9, 5, 5, 2, 4, 5, 3, 7, 5, 1, 4, 6, 9, 
9, 1, 5, 6, 5, 2, 8, 5, 2, 3, 5, 1, 9, 3, 2, 0, 1, 0, 1, 5, 0, 3, 7, 5, 
3, 0, 4, 1, 1, 8, 2, 0, 1, 0, 2, 8, 2, 6, 5, 1, 4, 8, 7, 2, 0, 0, 7, 3, 
7, 9, 9, 1, 6, 0, 2, 2, 3, 8, 8, 2, 7, 4, 1, 5, 5, 1, 8, 1, 0, 8, 4, 1, 
9, 2, 7, 8, 2, 5, 1, 0, 5, 9, 7, 2, 6, 2, 3}

{9, 6, 5, 9, 0, 6, 0, 8, 5, 7, 6, 2, 1, 5, 9, 2, 1, 2, 1, 5, 7, 0, 6, 2, 
3, 7, 0, 5, 5, 0, 4, 5, 2, 0, 4, 0, 8, 5, 7, 2, 6, 8, 1, 3, 3, 6, 5, 0, 
9, 7, 4, 5, 8, 9, 2, 5, 6, 2, 9, 6, 6, 3, 9, 4, 9, 2, 7, 3, 7, 7, 0, 2, 
4, 9, 0, 0, 7, 5, 7, 3, 0, 9, 3, 2, 7, 1, 1, 1, 0, 7, 7, 7, 1, 6, 8, 8, 
1, 1, 1, 4, 7, 4, 8, 5, 0, 8, 2, 7, 0, 5, 1}

{9, 8, 3, 4, 3, 9, 7, 8, 0, 5, 8, 6, 9, 2, 8, 2, 3, 5, 1, 1, 4, 4, 8, 7, 
5, 5, 3, 5, 5, 4, 0, 1, 3, 6, 3, 5, 3, 7, 3, 1, 5, 4, 1, 6, 2, 8, 6, 8, 
8, 3, 1, 1, 2, 1, 9, 0, 3, 7, 6, 0, 4, 5, 0, 8, 1, 6, 2, 7, 9, 9, 7, 0, 
5, 5, 9, 7, 3, 9, 2, 0, 6, 7, 5, 7, 9, 3, 9, 7, 5, 0, 0, 4, 0, 2, 9, 4, 
1, 3, 8, 7, 5, 3, 4, 5, 5, 4, 4, 5, 5, 1, 6}

{9, 9, 1, 8, 7, 8, 4, 0, 7, 6, 5, 8, 2, 9, 4, 8, 1, 6, 9, 6, 6, 4, 2, 6, 
9, 2, 7, 7, 8, 4, 8, 8, 9, 3, 4, 0, 4, 8, 0, 4, 0, 6, 2, 9, 2, 0, 6, 6, 
3, 5, 4, 2, 2, 8, 7, 4, 5, 0, 5, 8, 3, 0, 7, 6, 1, 9, 5, 8, 1, 8, 4, 1, 
2, 5, 0, 3, 9, 1, 6, 8, 5, 9, 7, 3, 1, 0, 8, 1, 8, 9, 9, 1, 6, 2, 5, 0, 
8, 8, 3, 6, 7, 0, 7, 2, 8, 1, 3, 4, 9, 6, 9}

{9, 9, 5, 9, 9, 1, 2, 6, 5, 8, 9, 3, 4, 0, 4, 6, 5, 5, 1, 1, 9, 1, 9, 4, 
1, 3, 1, 7, 4, 5, 8, 2, 2, 9, 3, 0, 8, 5, 6, 6, 2, 2, 6, 6, 6, 2, 5, 0, 
3, 5, 5, 0, 4, 9, 7, 5, 3, 4, 3, 9, 9, 7, 1, 9, 6, 3, 6, 6, 5, 1, 6, 1, 
7, 2, 7, 3, 5, 1, 1, 6, 2, 3, 2, 8, 3, 3, 6, 4, 3, 9, 7, 7, 9, 9, 2, 7, 
6, 1, 3, 0, 0, 9, 7, 6, 6, 6, 0, 2, 2, 7, 8}

{9, 9, 8, 0, 1, 2, 8, 2, 6, 1, 7, 2, 9, 8, 2, 7, 8, 4, 1, 9, 0, 0, 3, 9, 
8, 1, 4, 5, 0, 8, 9, 6, 8, 5, 6, 5, 5, 3, 1, 4, 5, 2, 5, 3, 8, 6, 6, 4, 
3, 8, 9, 8, 4, 3, 3, 4, 7, 6, 2, 9, 4, 0, 3, 4, 9, 5, 1, 1, 7, 1, 7, 2, 
8, 6, 1, 2, 5, 7, 0, 6, 6, 4, 6, 6, 2, 2, 7, 4, 4, 2, 6, 4, 4, 6, 0, 9, 
0, 9, 8, 6, 6, 1, 1, 2, 2, 4, 6, 5, 2, 3, 8}

f(1) = A000038 </%7Enjas/sequences/A000038>

f(2) = A090986 </%7Enjas/sequences/A090986>

f(3) = {6, 6, 6, ..., 6, 6, 6, ...} = A010722 </%7Enjas/sequences/A010722>

Sincerely yours, Bob.

Richard Mathar wrote:

>Does it make sense to put infinite products of integer ratios involving
>small integer powers into the OEIS? These may look like a too small sample
>randomly chosen from a fuzzy set of other constants. The case of (n^2-1)/(n^2+1)
>is already A090986, and (n^3-1)/(n^3+1)=0.6666666.. roughly equivalent to A020793,
>therefore (n^4-1)/(n^4+1) is the smallest new value that comes into mind:
>
>
>  
>
.....................................................................................................................................................................

>Richard J. Mathar, www.strw.leidenuniv.nl/~mathar
>
>Digits := 120 :
>m := 1:
>for r from 2 to 10 do
>        omega := cos(Pi/r)+I*sin(Pi/r) :
>        x := (-1)^(m+1)*2*m*m!/r*mul( GAMMA(-m*omega^j)^(-(-1)^j),j=1..2*r-1) ;
>        x := Re(evalf(x)) ;
>        print(r,x) ;
>od:
>
>
>
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>  
>