Question
Let $(G,\cdot,e)$ be a non-commutative group and $s,a \in G$ .I'm looking for interesting functions $\chi_{s,a}:\Bbb N \rightarrow G$ witht this property
$$\chi_{s,a}(n)= \begin{cases} s, & \text{if $n$ is even} \\ a, & \text{if $n$ is odd} \end{cases}$$
My attempt for a closed form
I define the power of an element of $G$ as usual. then I tried to use the function $\theta(x)=\frac 12+\frac12(-1)^x$ and define $\chi$ as follow
$$\chi_{s,a}(n)=s^{\theta(n)}a^{\theta(n+1)}$$
Since $\theta(x)=\begin{cases} 1, & \text{if $n$ is even} \\ 0, & \text{if $n$ is odd} \end{cases}$ then $s^{\theta(n)}a^{\theta(n+1)}= \begin{cases} s^1a^0=se=s, & \text{if $n$ is even} \\ a^0a^1=ea=a, & \text{if $n$ is odd} \end{cases}$
Another partial attempt
If I define a function $F_a:\{0;1\}\times G\rightarrow G$ such that for every $x\in G$
$F_a(0,x)=x$ and $F_a(1,x)=a$ then $$\chi_{s,a}(n)=F_a(\theta(n),s)$$
But here I can't find a closed form for $F_a$ involving the group operation in an interesting way that is different from $F_a(q,x)=a^qx^{|q-1|}$
I'm looking for a big list of examples of nice alternative closed forms (involving the group operation too).