# Is there a simpler function $f$ equivalent to $f(x)=af(x-1)^2+bf(x-1)+c$?

I have been using the function $$f(x)=af(x-1)^2+bf(x-1)+c$$ for a project, but wanted to know if there was a closed form of the equation or a form of the function in relation to $$f(0)$$ or $$f(1)$$. If not, is there a way to solve for a, b, and c with 3 or 4 values of $$f(x)$$? I know that $$f(1)=af(0)^2+bf(0)+c$$ $$f(2)=a^3f(0)^4+2a^2bf(0)^3+2a^2cf(0)^2+ab^2f(0)^2+2abcf(0)+ac^2+abf(0)^2+b^2f(0)+bc+c$$ and so on, but it would be much easier if there was a simpler way of evaluating this equation.

• just think that f(x) could be $sin(x), e^x, x^2 , e^{sin(x)} you can not evaluate ist if you do not specify f May 20 at 20:09 • If$f(k)$is known for$k=0,1,2,3$then you have three linear equations$f(1)=af(0)^2+bf(0)+c$,$f(2)=af(1)^2+bf(1)+c$,$f(3)=af(2)^2+bf(2)+c$from which you can solve for$a,b,c$, assuming there is a solution. May 20 at 20:13 • By$f(x)=(g(x)-b/2)/a$you can convert it to$g(x)=g(x-1)^2+K$where$K=-b^2/4+ac+b/2$. I don't think there is a closed form possible for that, maybe in few special cases ... – Sil May 20 at 20:15 • What you ask is an equivalent of logistic map. May 21 at 9:51 • The OEIS sequence A003095 is a simple example where$\,a=1,b=0,c=1\,\$ with lots of references. May 25 at 0:53