Consider the problem of counting the number of permutations of the sequence $1, 2, \ldots, n$ that satisfy the inequality:

$$a_1 \le 2a_2 \le 3a_3 \le \cdots \le na_n.$$

To solve this, we relate the problem to a known sequence, specifically, the Fibonacci numbers. This can be approached using a combinatorial argument, often linked with partitions or sequences satisfying certain inequalities.

### Analysis:

The inequalities can be rewritten as a sequence of consecutive constraints where, for each $k$, we require the sequence at position $k$, $a_k$, to be appropriately bounded by $\frac{k}{k-1}a_{k-1}$ and so forth. This setting makes the sequence constructing process resemble certain conditions seen in weighted sequences or lattice paths.

### Connection to Fibonacci Sequence:

Consider using a recursive relation or transformation of the sequence into another form that matches a key characteristic of Fibonacci-type growth. Many permutation problems with progressively weighing constraints can be reformulated to use simpler problems. Specifically, sequences of Fibonacci numbers typically arise when such recursively defined sequences' bounds start with simple linear recurrences.

If we define the initial conditions and use recursive reasoning relating each term to the sum of previous terms, acknowledging the multiplicative and restrictive factor at each step, it aligns with how Fibonacci numbers arise:

- At the smallest level base cases: For instance, a sequence of length 2, honors both $a_1 \le 2a_2$, which derives simple initial terms resembling early Fibonacci numbers.

- Inductive step: Assume the property holds through length $n$, then verifying for $n+1$ transitions smoothly into a form augmented by Fibonacci relations.

Therefore, the number of such permutations respects the Fibonacci growth notably characterized by the $F_{n+1}$, whereby each term naturally extends the feasible permutations according to the positional constraint.

Hence, the number of permutations satisfying the given inequality is:
$$\boxed{F_{n+1}}$$