A solution has been found for this problem that does not involve convoluted sequences. Thanks to anyone who consider it, for however briefly they did.
Is it possible to find a closed form to a single variable sequence with a multivariable recurrence relation?
For example, define by where if is even and if is odd. Note: is a bijection. Next, let be a sequence with the following recurrence relations:
and the extremely nonlinear relation:
For this question, assume that and through are known where they are the minimum number of elements needed to start the recursion.