# tetris tiling 2xn or ' filling a tube '

In how many ways can a rectangle $$2$$ units wide and $$8$$ units high be tiled with Tetris blocks? Extra credit: Let n be an even integer. In how many ways can a rectangle $$2$$ units wide and $$n$$ units high be tiled with Tetris blocks? I found a recursion that leads to $$F_{n-1}F_n$$ Fibonacci but still confused as to whether I found all configurations of each tetromino. ruling out the 'S' and 'T' piece we only need to find the number of ways the square, straight $$4$$, and L piece can fit into the $$2\times n$$ 'tube' right? someone always helps on here, thanks in advance.

• The S and T pieces can fit into the tube if oriented correctly, so why do you leavce them out? – Michael Lugo Dec 4 '18 at 16:26
• because an S piece will leave holes as well as the T pieces, try drawing it .a 2 square wide 'tube' , not the standard Tetris Tm game area. – Randin Dec 4 '18 at 17:19
• That makes sense. For others who don't see it: the area below an S or T piece is odd, but we need the area to be a multiple of 4 in order to fill it in. – Michael Lugo Dec 4 '18 at 20:39