# Nonlinear recurrence relation

• Sep 1st 2012, 08:31 PM
SlipEternal
Nonlinear recurrence relation
Is it possible to find a closed form to a single variable sequence with a multivariable recurrence relation?

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

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

and the extremely nonlinear relation:

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

For this question, assume that $\displaystyle \min{\mathbb{N}} = 1$ and $\displaystyle a_1$ through $\displaystyle a_i$ are known where they are the minimum number of elements needed to start the recursion.
• Sep 2nd 2012, 10:10 AM
SlipEternal
Re: Nonlinear recurrence relation
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.