DE-BUNDESWETTBEWERB 2002 Problem 6

We consider the sequences strictely increasing $(a_0,a_1,...)$ of naturals which have the following property : For every natural $n$ , there is exactly one representation of $n$ as $a_i+2a_j+4a_k$ , where $i,j,k$ can be equal.

Prove that there is exactly a such sequence and find $a_{2002}$