Question 3
Let $n,m$ be positive integers with $m\gt n\ge5$, with $m$ depending on $n$. Consider the sequence $a_1,a_2,\ldots,a_m$, where
$$a_i=i\quad(1\le i\le n),$$
$$a_{n+j}=a_{3j}+a_{3j-1}+a_{3j-2}\quad(1\le j\le m-n),$$
with $m-3(m-n)=1$ or $2$, that is,
$$a_m=a_{m-k}+a_{m-k-1}+a_{m-k-2},\qquad k=1\text{ or }2.$$
Thus, if $n=5$, the sequence is $1,2,3,4,5,6,15$; if $n=8$, it is $1,2,3,4,5,6,7,8,6,15,21$. Find $S=a_1+a_2+\cdots+a_m$ if (i) $n=2007$ and (ii) $n=2008$.
$$a_i=i\quad(1\le i\le n),$$
$$a_{n+j}=a_{3j}+a_{3j-1}+a_{3j-2}\quad(1\le j\le m-n),$$
with $m-3(m-n)=1$ or $2$, that is,
$$a_m=a_{m-k}+a_{m-k-1}+a_{m-k-2},\qquad k=1\text{ or }2.$$
Thus, if $n=5$, the sequence is $1,2,3,4,5,6,15$; if $n=8$, it is $1,2,3,4,5,6,7,8,6,15,21$. Find $S=a_1+a_2+\cdots+a_m$ if (i) $n=2007$ and (ii) $n=2008$.