# How to solve this recurrence equation with Mathematica?

I am interested in solving this recurrence relation (equation) with Mathematica code. I am new to Mathematica. I tried RSolve, RSolveValue, AsymptoticRSolveValue commands for this. But I am not getting any result for different values of 'theta'. Any help would be highly appreciated. Thanks in advance.

RSolve[{a[n + 1] == \[Theta]*a[n] (-1 + a[n]), a[0] == 1}, a[n], n]

• A nonlinear recurrence is not likely to have an explicit solution known to Mathematica. Of course, you can still use RecurrenceTable[] or the method in Akku's solution to generate the terms. Jul 8 '20 at 12:32
• Asymptotic solutions are a[n] = 0 and a[n] = 4/3. Jul 9 '20 at 1:51
• Thank you for the suggestion. Jul 10 '20 at 18:31

Define a recursion relation (with memorizing former values, which speeds up) like

a[0] = a0;
a[n_] := a[n] = \[Theta]*a[n - 1] (-1 + a[n - 1])

a[3]

(*   (-1 + a0) a0 \[Theta]^3 (-1 + (-1 + a0) a0 \[Theta]) (-1 + (-1 +
a0) a0 \[Theta]^2 (-1 + (-1 + a0) a0 \[Theta]))   *)


But with a[0]=1, you get zero for n>0.

• Thank you very much. Jul 10 '20 at 18:29