Tilings math challenge

[enthumath]

Isabel was making a rectangular strips 2 units wide using 2 x1 tiles. She was interested in the number of different strips of each length she could make. Here are some of her examples arranged according to units of length. Isabel regards all the strips shown as different even though some are flips of others.



a) draw the eight ways of making strips 5 units long- (i have already solved this)

Isabel noticed that the number of ways of making strips followed a pattern: the number of strips n units long is always the sum of the numbers of the strips of the two previuos lengths. By way of example, if Wn is the number of strips of length n the diagrams above show that W4= W3 + W2 since 5= 3+2. In general her guess is that

W1= 1, W2= 2 and W1= W(N-1) + W(N-2) when n>or= 3

b) explain why Isabel's huess is true
c) Isabel experiemnted with 3 x 1 tiles, making strips 3 units wide. Find a formula similar to that above, for the number of strips of length n

d) Find a similar formula for the number of ways of making strips of width m and length n from mx1 tiles

I would appreciate your help- Thank you very much