Question

find and solve a recurrence relation for the number of ways to arrange flags on an n-foot flagpole using three types of flags: red flags 2 feet high, yellow flags 1 foot high, and blue flags 1 foot high. page 4 of 8

Answer 1

Let Fn be the number of ways of arranging such flagpole with the given conditions.

When arranging a flagpole of n feet high, consider the following cases

If the last flag used is a red flag, then the other flags are n-1 foot high, so they can be seen as arranged on a smaller flagpole of n-1 feet high, which can be done in Fn-1 ways.

Similarly, If the last flag used is a gold flag, then the other flags can be seen as arranged on a smaller flagpole of n-1 feet high. This can be done in Fn-1 ways.

If the last flag used is green, the other flags are n-2 feet high, so the flagpole can be arranged in Fn-2 ways.

Using the sum rule, we obtain that Fn = 2Fn-1 + Fn-2 for all n≥3. Listing all the combinations of flags, the initial conditions are F1 = 2 and F2 = 3.

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