My data doubles, then triples, then doubles, then triples. I can't figure out how to write a function $f(N)$ for the data without using a piecewise function. Is it possible?
Here is an example if the first datum was $1$:
$$f(1)=1\qquad f(2)=2\qquad f(3)=6\qquad f(4)=12\qquad f(5)=36\qquad f(6)=72$$
So essentially you want to have $f(2n+1)=c\cdot 2^n\cdot 3^n$ and $f(2n+2)=c\cdot 2^{n+1}\cdot 3^n$ where $c=f(1)$ without dividing into cases like this? Perhaps $$ f(N)=c\cdot 2^{\frac12\left(N-0.5+0.5\cdot(-1)^N\right)}\cdot 3^{\frac12\left(N-1.5-0.5\cdot(-1)^N\right)} $$ Here is a table made with Wolfram|Alpha showing how these exponents behave. And here is another table showing values of $f$ for $c=1$ corresponding to your example.