Equations:
${ F_{0}}={ F_{1}}+{ F_{3}}$
${ F_{1}}=1$
${ F_{3}}=-3{\frac { \left( {x}^{2}+1/3 \right) \left( x+1 \right) x}{x-1}}$

Coefficients:
$1, 1, 2, 5, 8, 8, 8, 8, 8, 8,\ldots$

Minimal polynomial:
$ \left( x-1 \right) F \left( x \right) +3{x}^{4}+3{x}^{3}+{x}^{2}+1$

Generating function:
$-{\frac {3{x}^{4}+3{x}^{3}+{x}^{2}+1}{x-1}}$

Recurrence relation:
$-a \left( n \right) +8$
$a \left( 0 \right) =1$
$a \left( 1 \right) =1$
$a \left( 2 \right) =2$
$a \left( 3 \right) =5$
$a \left( 4 \right) =8$

Closed form:
$\cases{1&$n=0$\cr 1&$n=1$\cr 2&$n=2$\cr 5&$n=3$\cr 8&otherwise\cr}$