For Exercises 49–51, we define inflation and deflation of a sequence of “tiles” consisting of s (“shorts”) and s (“longs”).

  • Inflation: Replace each by and each by . For example, the inflation of ()()() would be , where we have inserted parentheses for clarity.
  • Deflation: Replace each by and each lone by . For example, the deflation of ()()() would be , where we have inserted parentheses for clarity.

Question 20.79

49. (a) Start with just a single and repeat the inflation process, showing the stages, until you reach a stage with 21 tiles.

(b) How many tiles are there at each stage that you reached? (If you continue this process forever, you tile a half-line to the right; you could tile the entire line by reflecting this right half-line over to cover the left half-line. The result is called a Fibonacci tiling of the line because of the appearance of the Fibonacci sequence in the numbers of tiles.)

(c) If a line segment contains copies of the tile and copies of the tile, how many tiles will the inflation of the segment contain?

49.

(a)

(b) segments at stage

(c)