Sticks and stones problem Ten squares in a row are labelled 1, 2, 3, . . . 10, in order.  A counter starts at square 1.  At every step,
the counter can move ahead 1, 2, or 3 squares. However, beginning with the second step, the counter
cannot move the
same
number of squares as it did in the previous step. For example, in the first step,
the counter can move from square 1 to square 3. Then in the next step, the counter can move to square
4 or square 6, but not square 5.  Find the number of possible sequences of steps that take the counter
from square 1 to square 10. (In the last step, the counter must land exactly on square 10.)
Hi, 
I am thinking of using sticks and stones for this problem, but cannot think of a way to use it.  Can anyone help me?
Thanks!

I get $21$.
$2$ with no $3$'s:
$1,2,1,2,1,2$
$2,1,2,1,2,1$
$12$ with one $3$:
$3,1,2,1,2$
$3,2,1,2,1$
$1,3,2,1,2$
$2,3,1,2,1$
$1,2,3,1,2$
$1,2,3,2,1$
$2,1,3,1,2$
$2,1,3,2,1$
$1,2,1,3,2$
$2,1,2,3,1$
$1,2,1,2,3$
$2,1,2,1,3$
$7$ with two $3$'s:
$3,1,2,3$
$3,2,1,3$
$1,3,2,3$
$3,2,3,1$
$1,3,1,3,1$
$2,3,1,3$
$3,1,3,2$
$2+12+7=21.$
