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[answered] Let n 3 be an integer. Let Cn be a cycle of length n with v


. Let n ? 3 be an integer. Let Cn be a cycle of length n with vertices labelled v1, v2, . . . , vn in order. Let an be the numberof ways to colour the vertices of Cn using the colours {1, 2, 3}. For example, a3 = 6 and a4 = 18 (an illustration isposted on Learn).

a) Give a combinatorial proof that for any n ? 5, an = an?1 + 2an?2.(Hint: Partition all possible colourings of Cn into two cases. Use contractions.)

b)?Determine an explicit formula for an in terms of n for all n ? 3.(Note: You only need to write the main steps in the calculation.)

 


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