I want to formalize an idea of a function where each $x$ distance traveled by the function divides into the prior distance leaving a remainder that is always a power of a constant.
For instance, if we take this constant to be 1 half, and the distance to be $S$, then $S_1/S_2=1+0.5^{1}$, and $S_2/S_3=1+0.5^{2}$, and so on, depending on how you index it and such. (at $S_1$, $y=1$, at $S_2$, $y=2$, and onward.)
For clarity, it seems that, $S_{1.5}/S_{2.5}$ ought to equal $1+0.5^{1.5}$
I want the function to be simple and continuous but have not had luck finding it. So, I'd be happy for inputs.