So I was looking through Youtube to see if there were, yet again, any math equations that I thought that I might like to solve when I came across this video by Michael Penn that was apparently "Erdős but simpler"[$1$] with this as the question:
Given$$\begin{cases}a_0=&1\\a_n=&a_\left\lfloor\frac n2\right\rfloor+a_\left\lfloor \frac n3\right\rfloor\end{cases}$$$$\lim_{n\to\infty}\dfrac{a_n}n=\,?$$which I thought that I might be able to solve. Here is my attempt at doing so:
$$\begin{align}a_1&=a_0+a_0=1+1=2\\a_2&=a_1+a_0=2+1=3\\a_3&=a_1+a_1=2+2=4\\a_4&=a_2+a_1=3+2=5\\a_5&=a_2+a_1=3+2=5\\a_6&=a_3+a_2=4+3=7\\a_7&=a_3+a_2=4+3=7\\a_8&=a_4+a_2=5+3=8\\a_9&=a_4+a_3=5+4=9\\a_{10}&=a_5+a_3=5+4=9\\a_{11}&=a_5+a_3=5+4=9\\a_{12}&=a_6+a_4=7+5=12\end{align}$$Now I'll let $g_n=\dfrac{a_n}n$ and see what happens when we calculate $g_n$ for each of these terms that I have already calculated:$$\begin{align}g_1&=2\\g_2&=1.5\\g_3&=\dfrac43\\g_4&=1.25\\g_5&=1\\g_6&=\dfrac76\\g_7&=1\\g_8&=1\end{align}$$Now here's where I'm stuck. I tried plugging the points that I got into Desmos[$2$] to get an equation of some sort so I could approximate an equation that I then could use to get an answer that I could use to determine the limit of $\dfrac{a_n}n$, but it doesn't work. Here is a table of the strategies that I used and why they don't work (explanations that are too long can be found the "Notes" section of this question):
| What I have tried | What I get | Why it doesn't work |
|---|---|---|
| Logarithmic regression: $y_1\sim a+b\ln(cx_1+d)$ | $R^2=0.4439$ with $a=9.18093$, $b=-2.05547$, $c=1.3042$, $d=42.1349$ | [$3$] |
| Sinusoidal regression (with $\sin$): $y_1\sim a+b\sin(cx_1+d)$ | $R^2=0.5272$ with $a=1.16481$, $b=0.287905$, $c=0.389035$, $d=1.0123$ | With $\cos$, as we add more terms, $R^2\to0$ |
| Sinusoidal regression (with $\cos$): $y_1\sim a+b\cos(cx_1+d)$ | $R^2=0.5272$ with $a=1.16481$, $b=0.287905$, $c=0.389035$, $d=-0.5588498$ | See above |
| Exponential regression: $y_1\sim ab^{x_1}+c$ | $R^2=0.4447$ with $a=1.787828$, $b=0.963084$, $c=-0.288955$ | Again, as more terms are added, $R^2\to0$ |
Now, if that's the case, what can we do? I actually tried just plotting terms of $a_n$, but that didn't really seem to work. So, at this point, I decided to just give up.
My question
Is there any way that I might be able to solve this problem in any way?[$4$]
Notes
[$1$]The original question (since Michael Penn had modified it) was: Given the exact same system of equations defining the recursive formula (as noted in the beginning), show that $\lim_{n\to\infty}\frac{a_n}n=\frac{12}{\ln(432)}$." I had to go into the video to find this, but I wouldn't say it's cheating, since all I'm doing here is getting the original question for reference.
[$3$]The reason logarithmic regression seems to not be accurate here because the terms are constantly expanding, while this just eventually drops to $0$
[$4$]As stated in Note $1$, I'm supposed to get $\frac{12}{\ln(432)}$, however I guess my question is more "How would I achieve said answer if I didn't know that the solution was said answer?" now that I think about it.