The sequence $$ \left(r_{k}\right)_{k=0}^{\infty}=(1, 1, 3, 5, 9, 15, 25, 39, 61, 93,\cdots) $$ can be found at OEIS as sequence A207641 and is related to Ramanujan theta functions.
Unfortunately there is no explicit formula for $r_{k}$ or I wasn't able to see it.
So I'm asking if there exists such expression and if so write it here or point to the appropriate sources.
Thanks in advance.