[seqfan] Re: An equivalence for integer sequences (with more questions than answers)

[Giovanni Resta]

Paolo Lava wrote:

> To avoid “NA” we could consider the terms of the sequence as coefficients of a simple continued fraction. For instance:
> 
> a(n)	                N	        
> n	            .6977746580…	Positive integers
> 2*n	            .4463899659…	Even numbers
> 2*n-1	            .7615941560…	Odd numbers

The fact that periodic continued fractions like those
for 1,2,1,2,1,2 or 2,1,2,1,2 lead to numbers
like 0.733 = sqrt(3)-1  and 0.366=(sqrt(3)-1)/2 is well known.

Instead, I was a little surprised (given my immense ignorance...) by
the continued fractions for
1,2,3,4,5,6,... (naturals)  -> BesselI(1,2)/BesselI(0,2),
2,4,6,8,10,... (even numbers) -> BesselI(1,1)/BesselI(0,1)
1,3,5,7,9,... (odd numbers) -> tanh(1).

I confess I have no idea of "why" (I used the Inverse Symb. Calc.
to spot the identities and Mathematica to check them (numerically)
up to 50 digits.

giovanni