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In attempting to answer this question, I reduced it to a seemingly simple generating functions question, but after days of work was unable to construct a proof. Since I do not have experience trying to do asymptotics with generating functions, I would like to know if a proof is salvageable from these methods.

The problem introduces the sequence $a_n$, defined by $a_0 = 1$ and $$ a_{n}=a_{\left\lfloor n/2\right\rfloor}+a_{\left\lfloor n/3 \right\rfloor}+a_{\left\lfloor n/6\right\rfloor} $$

and asks for a proof that $$ \lim_{n\to\infty}\dfrac{a_{n}}{n}=\dfrac{12}{\log{432}} $$ Writing the generating function $\displaystyle A(x) = \sum_{n \ge 0} a_n x^n$, this translates to $$ A(x) = (1 + x)A(x^2) + (1 + x + x^2) A(x^3) + (1 + x + x^2 + \cdots + x^5)A(x^6) - 2 $$

Even better, let $b_0 = a_0$ and $b_n = a_n - a_{n-1}$ for all $n \ge 1$, and define the generating function $\displaystyle B(x) = \sum_{n \ge 0} b_n x^n = (1 - x)A(x)$. Multiplying the above by $(1-x)$ gives $$ (1 - x)A(x) = (1 - x^2)A(x^2) + (1 - x^3)A(x^3) + (1 - x^6)A(x^6) + 2x - 2 $$ i.e. $$ B(x) = B(x^2) + B(x^3) + B(x^6) + 2x - 2 \tag{1} $$

This formula is so elegant that it almost seems there has to be a solution involving it. After unsuccessfully trying to do assymptotics with the formula, I used it to find an explicit representation of $B$, using the Delannoy Numbers:

$$ B(x) = 1 + 2 \sum_{l, m \ge 0} \sum_{d \ge 0} 2^d {l \choose d}{m \choose d} x^{2^l 3^m} $$

It follows that in fact $$ a_n = 1 + 2 \sum_{d \ge 0} 2^d \sum_{l,m \ge 0}_{2^l 3^m \le n} {l \choose d} {m \choose d} \tag{2} $$

One can do naive bounds on the sum in (2) - replacing the condition $2^l 3^m \le n$ with $2^l 2^m \le n$ and $3^l 3^m \le n$ for upper and lower bounds, respectively. But this isn't good enough; it gives (after algebra and combinatorial work) approximately $$ \frac{n^{\log_3(1 + \sqrt{2}) - 1}}{2} < \frac{a_n}{n} < \frac{n^{\log_2(1 + \sqrt{2}) - 1}}{2} $$

This strongly suggests trying to approximate (2) with the condition $(1 + \sqrt{2})^l (1 + \sqrt{2})^m \le n$, but I have no idea how to justify that.

At any rate, I've made too much of what feels like progress to give up on the problem, and if anyone can think of a way to use (2) to get a solution or else to use (1) and find the asymptotics directly, I'd be very thankful.

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I recommend Dirichlet series instead of power series. Dirichlet series with coefficients $b_n$ is easy to find, then Wiener-Ikehara Tauberian Theorem solves the problem. This idea was already discussed in the AOPS forum from the link. –  i707107 Jan 9 at 13:24
    
But it will be nice to see if anyone solves directly with your formula for $b_n$. –  i707107 Jan 9 at 13:27

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