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I really want to know how I can find the suitable initial condition of a recurrence relation.

A binary message is transmitted over a channel using only two signals 0 and 1. Suppose that signal 0 requires one unit of time to transmit and signal 1 requires two units of time to transmit. Let n g be the number of possible messages that can be transmitted in exactly n units of time. Find a recurrence relation, with initial conditions, of the sequence G1, G2, ......

In this case, how to find the recurrence relation and the initial condition?

newDeveloper
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1 Answers1

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The recurrence $p_n=p_{n-1}+p_{n-2}$ with $p_1=1$ and $p_2=2$ solve the problem.

See that if a message has lenght $n$ then:

1) if the last sign has length $1$un then we get $p_{n-1}$ possibilities;

2) if the last sing has length $2$un then we get $p_{n-2}$ possibilities.

So,

$$p_n=p_{n-1}+p_{n-2}$$

If the message has length $1$ un then we have just $1$ possibility wich is signal $0$, so $p_1=1$.

If the message has length $2$ un then we have $2$ possibilites wich are: message 1:signal $0$ and signal $0$; and message 2:signal $1$. So $p_2=2$.

Arnaldo
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