RSolve does not reduce for a certain recurrence ratio

I'm trying to use RSolve in the following way to solve a recurrence relationship:

RSolve[{a[0] == 1, a[2 n + 1] == a[2 n]* 2, a[2 n] == a[2 n - 1] + 2}, a, n]

I think the meaning of the relationship is clear: for odd terms in the sequence, multiply the previous term by two, and for even terms, add two to the previous term.

However, when evaluating this, Mathematica simply echoes the entry:

Result of the evaluation

Instead of trying to solve the recurrence.

I can not find anything in the RSolve documentation that talks about cases in which RSolve will not do anything, without any error message.

Have I committed a syntax error or does this mean that Mathematica can not resolve this type of recurrence relationship? How can I change my entry for Mathematica to solve the recurrence, assuming it is possible?