Tatami are rectangular mats, used to completely cover the floor of a room, without overlap.

Assuming that the only type of available tatami has dimensions $1\times 2$, there are obviously some limitations for the shape and size of the rooms that can be covered.

For this problem, we consider only rectangular rooms with integer dimensions $a,b$ and even size $s=a\cdot b$.
We use the term 'size' to denote the floor surface area of the room, and — without loss of generality — we add the condition $a\le b$.

There is one rule to follow when laying out tatami: there must be no points where corners of four different mats meet.
For example, consider the two arrangements below for a $4\times 4$ room:

The arrangement on the left is acceptable, whereas the one on the right is not: a red "X" in the middle, marks the point where four tatami meet.

Because of this rule, certain even-sized rooms cannot be covered with tatami: we call them tatami-free rooms.
Further, we define $T(s)$ as the number of tatami-free rooms of size $s$.

The smallest tatami-free room has size $s=70$ and dimensions $7\times 10$.
All the other rooms of size $s=70$ can be covered with tatami; they are: $1\times 70$, $2\times 35$ and $5\times 14$.
Hence, $T(70)=1$.

Similarly, we can verify that $T(1320)=5$ because there are exactly $5$ tatami-free rooms of size $s=1320$:
$20\times 66$, $22\times 60$, $24\times 55$, $30\times 44$ and $33\times 40$.
In fact, $s=1320$ is the smallest room-size $s$ for which $T(s)=5$.

Find the smallest room-size $s$ for which $T(s)=200$.