n-dimensional colour building

This problem involves combining lengths using rods of two different colors.

Here is the problem link: https://nrich.maths.org/4338

And the solution link: https://nrich.maths.org/4338/solution

Read-only if you have explored the problem well otherwise it won't make sense:- SInce this is proven for the availability of rods till $k$ and for rods length $n$ , I was thinking of extending this to 2 dimensions. So now finding the possibilities of arrangements for rectangle $m \times n$ till the availability of rods $k$

Extension: Maybe even till nth dimension if we recognize a pattern. I would love to hear your views upon it :)

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$