A song about sequences

Algebra Level pending

Let \( (a_n)_{n \in \mathbb{N}} \) be a monotone increasing sequences of non negative integers. The sequence has the property, that each non-negative integer has a unique representation in the form of \( a_i + 2*a_j + 4*a_k \)

\(a_i , a_j, a_k \) don't have to be distinct.

Let x be the number of sequences that suffice the given property above.

Compute \( a_{2002} + x \)


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