Consider the following sequence defined by this recursion :
Let f(1) = f(2) = 1 and for n > 2
f(n) = [ f(n-1)^2 ( f(n-1) +1)^2 ] / [ f(n-2) ( f(n-2) + 1)^3 ]
Now we have that f(3) = 1/2 , f(4) = 9/128 , ...
and the sequence goes down all the time ( for every n : f(n+1) < f(n) ) until we get to 0 in the limit.
f( oo ) = 0
Somewhat surprisingly this goes down at an exponential rate.
We have
lim n to +infinity f(n+1) / f(n) = C
For a constant C.
C = 0.003879..
Me and my friend Mick have tried to find a (proven) closed form expression for this constant C.
But without succes.
see the unanswered question (unanswered at least till 14/10/2024) on mathstackexchange :
https://math.stackexchange.com/questions/4955849/limit-related-to-fn-fracfn-12-fn-112fn-2fn-213
and a related question :
https://math.stackexchange.com/questions/4956849/period-t-such-that-fnt-fn-and-fn-fracfn-112-fn-122