Nonlinear recurrence relation

Is it possible to find a closed form to a single variable sequence with a multivariable recurrence relation?

For example, define $f:\mathbb{N}\times\mathbb{N} \to \mathbb{N}$ by $f(i,j) = \frac{2^{i-1}(6j-5+4[i]_2)+1}{3}$ where $[i]_2 = 0$ if $i$ is even and $[i]_2 = 1$ if $i$ is odd. Note: $f$ is a bijection. Next, let $a_n$ be a sequence with the following recurrence relations:

$a_{4n-1} = 4a_n + 1$

and the extremely nonlinear relation:

$a_{f(i,j+1)} = 64a_{f(i,j)}+7(2^i+3)$

For this question, assume that $\min{\mathbb{N}} = 1$ and $a_1$ through $a_i$ are known where they are the minimum number of elements needed to start the recursion.