I found this sequence: $$1,3,3,15,5,35,7,63,9,99,11,143,13,\ldots$$
I'm looking for its recurrence relation, and/or its closed form.
My take: It's easy to see that when $n\geq4$ and is even, $a_n=a_{n-1}\cdot a_{n+1}$, and when $n\geq3$ and odd, then $a_n=n$. I'm having trouble to "combine" these observations into one recurrence relation.
Add: was able to write the recurrence relation as: $$b_n=n^2-1+(-1)^n(b_{n-1}+(-1)^{n-1}(n-1)).$$ For $b_1=1$, though the resulting closed form doesn't look elementary. The generating function, using a CAS, is: $$\frac{x^5+x^4-6 x^3-3 x-1}{\left(x^2-1\right)^3}.$$