jmc

We proceed recursively. Suppose we can build $T_m$ towers using blocks of size $1,2,\dots ,m$. How many towers can we build using blocks of size $1,2,\dots ,m,m+1$? If we remove the block of size $m+1$ from such a tower (keeping all other blocks in order), we get a valid tower using blocks $1,2,\dots ,m$. Given a tower using blocks $1,2,\dots ,m$ (with $m\ge 2$), we can insert the block of size $m+1$ in exactly 3 places: at the beginning, immediately following the block of size $m-1$ or immediately following the block of size $m$. Thus, there are 3 times as many towers using blocks of size $1,2,\dots ,m,m+1$ as there are towers using only $1,2,\dots ,m$. There are 2 towers which use blocks $1,2$, so there are $2\cdot 3^6=1458$ towers using blocks $1,2,\dots ,8$, so the answer is $\boxed{458}$.