Los primeros términos calculados son : $1,1,1,2,3,4,10,18,28,80,162,280,880$
Así que la fórmula cerrada parece ser $\bbox[5px,border:2px solid]{\begin{cases}a_0=1\\a_1=1\\a_2=1\\a_n=(n-1)a_{n−3}\quad n\ge3\end{cases}}$
Veamos cómo es el determinante bajo la hipótesis de recurrencia hasta $a_{n+7}$ .
$D=\begin{array}{l} +a_{n+8}a_{n+4}a_{n}\\ +a_{n+5}a_{n+1}a_{n+6}\\ +a_{n+2}a_{n+3}a_{n+7}\\ -a_{n+6}a_{n+4}a_{n+2}\\ -a_{n+7}a_{n+5}a_{n}\\ -a_{n+8}a_{n+1}a_{n+3}\end{array} = \begin{array}{l} +(n+3)a_{n+8}a_{n+1}a_{n} \\ +(n+4)(n+5)(n+2)a_{n+2}a_{n+1}a_{n} \\ +(n+2)(n+6)(n+3)a_{n+2}a_{n}a_{n+1} \\ -(n+5)(n+2)(n+3)a_{n}a_{n+1}a_{n+2} \\ -(n+6)(n+3)(n+4)a_{n+1}a_{n+2}a_{n} \\ -(n+7)(n+4)(n+2)a_{n+2}a_{n+1}a_{n} \end{array}$
$=(n+3)a_{n+8}a_{n+1}a_{n}+[-n^3-14n^2-61n-82]\,a_{n+2}a_{n+1}a_{n}$
$=(n+3)a_{n+8}a_{n+1}a_{n}-[(n+3)(n+7)(n+4)-2]\,a_{n+2}a_{n+1}a_{n}$
$=(n+3)a_{n+1}a_{n}\times[a_{n+8}-(n+7)(n+4)a_{n+2}]+2a_{n+2}a_{n+1}a_{n}$
$=(n+3)a_{n+1}a_{n}\times[a_{n+8}-(n+7)a_{n+5}]+2a_{n+2}a_{n+1}a_{n}$
Examinemos para $n=3k,\ 3k+1,\ 3k+2$ el producto $a_{n+2}a_{n+1}a_{n}$
$a_{n+2}a_{n+1}a_{n}=(n+1)(n)(n-1)a_{n-1}a_{n-2}a_{n-3}=\ldots=(n+1)(n+0)(n-1)\ldots(4)(3)(2)a_2a_1a_0=(n+1)!\times 1\times 1\times 1=(n+1)!$
$a_{n+2}a_{n+1}a_{n}=(n+1)(n+0)(n-1)\ldots(5)(4)(3)a_3a_2a_1=\frac{(n+1)!}{2}\times 2\times 1\times 1=(n+1)!$
$a_{n+2}a_{n+1}a_{n}=(n+1)(n+0)(n-1)\ldots(6)(5)(4)a_4a_3a_2=\frac{(n+1)!}{2\times 3}\times 3\times 2\times 1=(n+1)!$
Así que $D=(n+3)a_{n+1}a_{n}\times[a_{n+8}-(n+7)a_{n+5}]+2\,(n+1)!$
Esto permite concluir que $a_{n+8}-(n+7)a_{n+5}=0$ que pone fin a la recurrencia.