Optimal algorithm for signaling for help?
Your problem is very similar to the well-known cow-path problem, though the details are a bit different. When talking about optimal algorithms, we are usually concerned with minimizing the competitive ratio (a concept from the field of online algorithms), which is the worst ratio between the solution produced by your algorithm and the one produced by the optimal algorithm (in this case, $n$). If your algorithm searches the sequence $a_1 &lt; a_2 &lt; \ldots$ then the competitive ratio is
$$ \sup_n \frac{\sum_{i\colon a_{i-1} &lt; n} a_i}{n}. $$
(Here $a_0 = 0$.)

Suppose we're aiming for a competitive ratio of $C$ (your algorithm gives $C = 4$). Considering $n=1,a_1+1,a_2+1,\ldots$, we must have
$$
\begin{align*}
a_1 &amp;\leq C, \\
a_1 + a_2 &amp;\leq C(a_1 + 1), \\
a_1 + a_2 + a_3 &amp;\leq C(a_2 + 1), \\
&amp;\ldots
\end{align*}
$$
The classical solution to the cow-path problem (Searching in the plane) shows that $C \geq 4$, and so your algorithm achieves the optimal competitive ratio.