Solve $f(x) = f(\lfloor x/2 \rfloor) + f(\lfloor x/3 \rfloor)$

Question

Solve the following recurrence relation $$f(1)=1, f(2)=2\\ f(x) = f\left(\left \lfloor \frac{x}{2} \right \rfloor\right) + f\left(\left\lfloor \frac{x}{3} \right\rfloor\right), \forall x \in \mathbb{N}, x \geq 3 $$

I tried the simple ways to solve a recurrence relation but got things messed up. Any Hint will be helpful.

Answer 1

Is $x$ constrained in some particular set?

Note that:

$$ \begin{aligned} f(2) &= f\left(\left\lfloor \frac{2}{2} \right\rfloor\right) + f\left(\left\lfloor \frac{2}{3} \right\rfloor\right)\\ &=f(1) + f(0) \end{aligned} $$

Therefore, $f(0) = f(2) - f(1) = 1$. At the same time, for $x=1$,

$$ \begin{aligned} f(1) &= f\left(\left\lfloor \frac{1}{2} \right\rfloor\right) + f\left(\left\lfloor \frac{1}{3} \right\rfloor\right)\\ &=2f(0), \end{aligned} $$

therefore, $f(x) = \tfrac{1}{2}f(1) = \tfrac{1}{2}$, which is a contradition. Therefore, $f$ cannot be defined (at least) at $x=0$.