파란띠 5수능형

수열들로 정의한 수열의 합

다음과 같이 세 수열이 정의된다.

(가) $a_{\scriptstyle 1}=1,\,a_{\scriptstyle 2}=2,\,a_{\scriptstyle 3}=4,\quad a_{\scriptstyle n}=2a_{\scriptstyle n-1}+a_{\scriptstyle n-2}-2a_{\scriptstyle n-3}\;\left(n>3\right)$
(나) $b_{\scriptstyle 1}=1,\,b_{\scriptstyle 2}=1,\,b_{\scriptstyle 3}=1,\quad b_{\scriptstyle n}=2b_{\scriptstyle n-1}+b_{\scriptstyle n-2}-2b_{\scriptstyle n-3}\;\left(n>3\right)$
(다) $c_{\scriptstyle 1}=-\sqrt{3},\,c_{\scriptstyle 2}=\sqrt{3},\,c_{\scriptstyle 3}=-\sqrt{3},\quad c_{\scriptstyle n}=2c_{\scriptstyle n-1}+c_{\scriptstyle n-2}-2c_{\scriptstyle n-3}\;\left(n>3\right)$

모든 항이 양수인 수열 $\left\{d_{\scriptstyle n}\right\}$ 이 $\vphantom{\begin{matrix}1\\1\end{matrix}}\displaystyle d_{\scriptstyle n}^{\scriptstyle 2}=\left(\sum_{\scriptstyle k=n}^{\scriptstyle n+2}a_{\scriptstyle k}^{\scriptstyle 2}\right)\left(\sum_{\scriptstyle k=n}^{\scriptstyle n+2}b_{\scriptstyle k}^{\scriptstyle 2}\right)\left(\sum_{\scriptstyle k=n}^{\scriptstyle n+2}c_{\scriptstyle k}^{\scriptstyle 2}\right)$ 일 때 $\vphantom{\begin{matrix}1\\1\end{matrix}}\displaystyle\dfrac{d_{\scriptstyle 1}}{405}\sum_{\scriptstyle n=1}^{\scriptstyle 8}d_{\scriptstyle n}$ 의 값은?