generating sequence with a specific recursive condition

Question

I'm looking for two different sequences that satisfy $x_n=x_{2n}+x_{2n+1}$ for all $n\geq 1$, whats a general possible way to find such sequences/ what other sequences satisfy that condition?

my idea: $x_n = \{0,0,0, \dots ,0\}$

and another im having trouble describing. it goes something like this:

$x_n = \{1, \frac{1}{2}, \frac{1}{2},\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},\underbrace{\frac{1}{8},\dots,\frac{1}{8}}_\text{8 of them}, \underbrace{\frac{1}{16},\dots,\frac{1}{16}}_\text{16 of them} \dots\}$

Answer 1

$x_n=x_{2n}+x_{2n+1}$ is a very loose condition; it leaves a great deal of freedom. Both your examples are fine, but there are more such sequences. In fact, there are immensely many.

In general, it goes like this:

1. Select an arbitrary value for $x_1$.
2. Select an arbitrary value for $x_2$.
3. Set $x_3$ to $x_1-x_2$.
4. Select an arbitrary value for $x_4$.
5. Set $x_5$ to $x_2-x_4$.
6. Select an arbitrary value for $x_6$.
7. Set $x_7$ to $x_3-x_6$.

$\dots$