Number of Power-Constrained Ternary Sequences

Discrete Mathematics Level 5

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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