Joyce Smith

2022-01-06

What is a recurrence relation for the number of sequences of yellow and purple stones of length N, where each sequence has the property that no two adjacent stones are purple.

Piosellisf

Beginner2022-01-07Added 40 answers

So,we have to find a recurrence relation for the number of sequences of yellow and purple stones of length $N$ , where, each sequence has the property that no two adjacent stones are purple.

Let$a}_{N$ be the number of such sequences of length $N$ .

There are two cases:

1) The last stone is yellow.

In this case, the first$n-1$ stones must not have 2 adjacent purples.

Thus, there are$a}_{N-1$ such sequences.

2)The last stone is purple.

In this case, the last 2 stones must be yellow and purple. So, the first$N-2$ stones must not have 2 adjacent purples.

Thus, there are$a}_{N-2$ such sequences.

In total, there are$a}_{N-1}+{a}_{N-2$ sequences.

Therefore,$a}_{N}={a}_{N-1}+{a}_{N-2$ , here, ${a}_{0}=1$ , ${a}_{1}=2$ . Hence, the required recurrence relation is $a}_{N}={a}_{N-1}+{a}_{N-2$

Let

There are two cases:

1) The last stone is yellow.

In this case, the first

Thus, there are

2)The last stone is purple.

In this case, the last 2 stones must be yellow and purple. So, the first

Thus, there are

In total, there are

Therefore,

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