[seqfan] Re: What constant is involved in creating this cf?

[DAN_CYN_J]

----- Original Message -----


From: "Olivier Gerard" <olivier.gerard at gmail.com> 
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu> 
Sent: Wednesday, July 24, 2013 5:38:46 AM 
Subject: [seqfan] Re: What constant is involved in creating this cf? 

On Wed, Jul 24, 2013 at 9:48 AM, <allouche at math.jussieu.fr> wrote: 

> Hi 
> 
> Expansions of powers of e and "combinations" of those are more 
> or less classical. See, e.g., 
> http://en.wikipedia.org/wiki/Continued_fraction 
> and more precisely 
> 
> http://en.wikipedia.org/wiki/Continued_fraction#Regular_patterns_in_continued_fractions 
> 
> We find there in particular 
> tanh(1/n) = [0 ; n, 3n , 5n, 7n, ...] 
> hence tanh(1) = [0 ; 1, 3, 5, 7, ...] 
> 

Dan, as Jean-Paul wrote, these ones are known since Euler, at least. 

I was wondering what your question was exactly. 

By adjusting with rationals and inversion, 
one can have the sequence starts where one wants. 

For instance 

1/tanh(1) = (e^2+1)/(e^2-1) gives [1; 3, 5, 7, 9, 11, ...] 
(e^2-1)/2  gives [3; 5, 7, 9, 11, ...] 
2/(e^2-7)  gives your [5; 7, 9, 11, 13, ...] 
(e^2-7)/(37-5e^2) gives [7; 9, 11, 13, 15, ...] 
(37-5e^2)/(36e^2-266) gives [7; 9, 11, 13, 15, ...] 

and you can go on as far as you want. 

Coefficients of the rational fractions are given by 

A001515 Bessel polynomial y_n(1) 

1, 2, 7, 37, 266, 2431, 27007, 353522, 5329837, 90960751, 1733584106, 

for the constant coefficient  (the link to CFs is mentionned by Benoit 
Cloitre 
in this sequence). 

and 
A000806 Bessel polynomial y_n(-1) 

1, 0, 1, -5, 36, -329, 3655, -47844, 721315, -12310199, 234615096 

for the coefficient of e^2 

and this sequence count among other things sets of pairs 
of integers without successive integers. 


Olivier 

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Thanks Olivier, 



About the different sequences that can be created from t e^2 

I did not know this. 

I  guess I just reinvented the wheel what I found with e^2. 

It is a fasinating  constant. 



Thanks , 



Dan