How to solve this functional (recurrence) equation ?

1. jk22

361
I'm in a problem where I have to solve the following functional equation :

$$F(n)^2=n+F(n+1)$$

Does anyone know some methods to solve this kind of problems ?

A similar equation happens in Ramanujan example of root denesting : http://en.wikipedia.org/wiki/Nested_radical#Square_roots

2. mathman

6,750
Don't know of a method. As stated the problem is incomplete - you need an initial condition (F(0) = ?).

3. jk22

361
The problem is to find F(1), knowing that $$F(1)=\sqrt{1+\sqrt{2+\sqrt{3+\ldots}}}$$.

4. Mentallic

3,783
Should that be to find F(n) given that $F(1)=\sqrt{1+\sqrt{2+\sqrt{3+\ldots}}}$ ?

Find F(2) and the pattern becomes clear.