Closed Forms For the Logistic Map

By means of trigonometric substitutions, it's not too difficult to
determine closed form solutions for the logistic map

               x[t+1]  =  A x[t] (1 - x[t]) 

for certain specific values of A.  For A=4 the solution is

     x_n  =  (1 - cos(2^n g))/2    where    g = invcos(1 - 2 x_0)

For A=2 the solution is

     x_n  =  (1 - exp(2^n g))/2    where    g = log(|1 - 2 x_0|)

For A=-2 the nth iterate can be expressed as

     x_n  =  1/2 - cos(pi/3 + (-2)^n g))

with the appropriate value of g.  By the way, the logistic formula
has a long history.  It was first used by the Belgian sociologist
and mathematician Pierre Francios Verhulst (1804-1849) to model
the growth of populations with limited resources.