Is it possible to transform following recurrence relation $a_n=4a_{n-2}-a_{n-4}$, $a_0=1$, $a_1=0$, $a_2=3$, $a_3=0$ so that it will have nonnegative coefficients? Number of terms, of course, can be be larger.
EDIT: Moving terms to left side is not that I actually wanted. I need that there is only one element on the left, with largest index.
For example: sequence $a_n=1-a_{n-1}$, $a_0=1$, $a_1=0$ is the same as $a_n=a_{n-2}$, $a_0=1$, $a_1=0$, but second one has only nonnegative coefficients.