What is the general term of this sequence?

[Denisse]

Could you help me to find the general term of the sequence:

$1 , \frac{5}{3} , 1 , \frac{15}{17} , 1 , \frac{37}{35} , 1 , \frac{63}{65} ,...$

Thank you!

 

[mfb]

There are infinitely many general expressions for series which begin like that.
If you look for the "easiest" expression, there could be something simple, but I don't see it at the moment.
The numerator seems to follow the pattern (same as denominator, 2 more, same, 2 less, same, 2 more, ...).
I don't see a clear pattern for the denominator, however.

 

[micromass]

I don't see a clear pattern for the denominator, however.

Think about perfect squares.

 

[Denisse]

Think about perfect squares.

Perfect squares? How? I don't see it

 

[MarneMath]

Well, 5/3 is (2^2 + 1)/(2^2 - 1) and 15/17 = (4^2 - 1)/(4^2+1) and 37/35 = ( 6^2 +1)/(6^2-1) and so on and so forth.

 

[mfb]

Think about perfect squares.

Oh, nice. I was too focused on powers of 2, which are close (+1, or +3 in one case) to all of the "visible" denominators.