Compositions with no 2

A composition of \( n \) is an expression of \( n \) as a sum of not necessarily distinct positive integers, where the order matters. Note that \( n = n \) counts as a composition of \( n \).

Let \( C_n \) be the number of compositions of \( n \) with no part equal to 2.

For instance, \( C_5 = 7 \) because \( 5=5=4+1=3+1+1=1+4=1+3+1\\=1+1+3=1+1+1+1+1\).

Find \( C_{15} \).


Inspiration.

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