Number of Power-Constrained Ternary Sequences

Consider a ternary sequence of length $$n$$, consisting of $$n$$ symbols, each taken from the set $$\{0,1,2\}$$. This sequence is to be transmitted over a power-constrained wireless link. Assume that the symbols $$0,1,2$$ takes $$0,1,2$$ unit of energy respectively, for transmission. Due to the power-constraint, only those sequences are eligible for transmission whose required average transmission energy is at most $$\frac{1}{2}$$.

In other words, for a sequence $$(s_1,s_2,\ldots, s_n)$$, if the $$i^\text{th}$$ transmitted symbol $$s_i$$ consumes $$w_i$$ amount of energy, then the sequence is eligible for transmission if and only if $\frac{1}{n} \sum_{i=1}^{n}w_i \leq \frac{1}{2}$ As an explicit example, with $$n=4$$, the sequences $$(0,0,0,0), (0,0,0,2), (1,1,0,0)$$ are eligible for transmission, but the sequence $$(0,0,1,2)$$ is not.

Let $$N(n)$$ denote the number of eligible sequences of length $$n$$. Define $l= \lim_{n\to \infty} \frac{1}{n} \ln(N(n)),$ if the above limit exists, otherwise, define $$l=-100$$.

Find $$\lfloor 10^4 l \rfloor$$.

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