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...

The $n$th such equation is $$\left(\tfrac{5\times 10^{n-1}-2}{3}\right)^3+(5\times 10^{n-1})^3+\left(\tfrac{10^n-1}{3}\right)^3=10^{3n-1}+\tfrac{2}{3}(10^{n-1}-1)10^{2n}+5\times 10^{2n-1}+\tfrac{10^n-1}{3},$$or with $x=10^n$ we can write it as $$\left(\tfrac{x-4}{6}\right)^3+\left(\tfrac{x}{2}\right)^3+\left(\tfrac{x-1}{3}\right)^3=\tfrac{x^3}{10}+\tfrac{2}{3}\left(\tfrac{x}{10}-1\right)x^2+\tfrac{x^2}{2}+\tfrac{x-1}{3},$$which we can verify algebraically (both sides are $\tfrac{(x-1)(x^2+2)}{6}$).
