<s>[INST] How to derive this sequence: $1^3+5^3+3^3=153,16^3+50^3+33^3=165033,166^3+500^3+333^3=166500333,\cdots$? I found it is interesting but I don't know how the R.H.S is coming from the L.H.S,
i.e, how to derive this sequence?
The sequence is as follows:
$$1^3+5^3+3^3=153$$
$$16^3+50^3+33^3=165033$$
$$166^3+500^3+333^3=166500333$$
$$1666^3+5000^3+3333^3=166650003333$$
$$.$$
$$.$$
$$.$$
$$\text{and so on}$$
So,any help please...
 [/INST] First some notations, let express the repunits $r_n=11\cdots 1=\dfrac{(10^n-1)}9$ in term of $k=10^n$
$\begin{cases} 
a_n&=16\cdots 6&=10^{n-1}+6\ r_{n-1}&=\frac{k-4}6\\
b_n&=50\cdots 0&=5\times 10^{n-1}&=\frac{k}2\\
c_n&=33\cdots 3&=3\,r_n&=\frac{k-1}3\end{cases}$
The sum of cubes 
$f(n)=a_n^3+b_n^3+c_n^3=\left(\frac{k-4}6\right)^3+\left(\frac k2\right)^3+\left(\frac{k-1}3\right)^3=\dfrac{k^3-k^2+2k-2}6=\dfrac{(k-1)(k^2+2)}6$
Can be expressed like this:
$$\boxed{f(n)=\dfrac{(10^n-1)(10^{2n}+2)}{6}}$$
But we are interested in the non-factored expression:
$\begin{align}f(n)&=\dfrac{k^3-k^2+2k-2}6\\&=\dfrac{k^3-4k^2+3k^2+2k-2}6\\&=\left(\dfrac{k^2-4k^2}6\right)+\left(\dfrac {3k^2}6\right)+\left(\dfrac{2k-2}6\right)=k^2\,a_n+k\,b_n+c_n=\overline{a_nb_nc_n}\end{align}$
Where
$$\overline{a_nb_nc_n}=\underbrace{16\cdots 6}_{a_n\times 10^{2n}}\;\underbrace{50\cdots 0}_{b_n\times 10^n}\;\underbrace{33\cdots 3}_{c_n}$$
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