A walk is made up of diagonal steps from left to right, starting at the origin and ending on the x-axis as in the example illustrated which shows 8 steps? How many paths are there for 4 steps, for 6 steps, for 8 steps? Draw the paths on a diagram. Compare your answers to the values of

${\displaystyle \frac{1}{2}(3^{\frac{n-2}{2}}+1)}$

where $n=2,4,6,8$. What about $n=10$?