Question 57
The $\pi$cchi are animals living in families of 1, 2 or 3 individuals. Originally a family of $3$ $\pi$cchi was imported into Italy. A family of $n$ $\pi$cchi reproduces by growing by $2n-2$ new individuals, forming a total of $3n-2$, and splitting into new families (two families with the same number of individuals do not necessarily split in the same way). All families reproduce simultaneously at each brood. For example, after the first brood there are necessarily seven $\pi$cchi, which could be divided into three families of two and one of one, or into one of three and two of two, or into seven of one, and so on. How many possibilities are there for the total number of $\pi$cchi after the seventh brood?
(A) 129
(B) 253
(C) $2^{7}$
(D) $\frac{2^{7}+3^{7}+1}{2}$
(E) $3^{7}$
(A) 129
(B) 253
(C) $2^{7}$
(D) $\frac{2^{7}+3^{7}+1}{2}$
(E) $3^{7}$