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I was asked by OEIS contributor Andrew Howroyd to post a Code Golf Challenge to extend [OEIS sequence A049021](https://oeis.org/A049021).
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> Would be super great to get a couple more terms for [...] A049021. Kind of thing [...] team golf would excel at.
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As I found out the hard way with the help of [user202729](https://codegolf.stackexchange.com/users/69850/user202729), the definition of A049021 is... *slippery*. Instead, this challenge will have you compute a similar sequence.
(If, separate from this challenge, you're interested in computing more terms of A049021, I encourage it!)
# Definition
This challenge will have you counting topologically distinct ways of partitioning a square into \$n\$ rectangles.
Two partitions are called *topologically distinct* if there's no [homeomorphism](https://en.wikipedia.org/wiki/Homeomorphism) \$A \rightarrow B\$ *with the additional property* that the corners of \$A\$'s square map surjectively to the corners of \$B\$'s square. The less fancy way of saying this is that if partitions \$A\$ and \$B\$ are topologically equivalent, there is a map from \$A \rightarrow B\$ and a map from \$B \rightarrow A\$ such that all nearby points stay nearby, and every corner of \$A\$ is sent to a distinct corner of \$B\$.
Examples of homeomorphisms are rotating the square, flipping the square, and "slightly" moving the boundaries of the rectangles.
When \$n = 4\$ there are seven topologically distinct partitions:
[![The seven topologically distinct ways of dissecting a square into 4 rectangles.](https://i.stack.imgur.com/rRQM8.png)](https://i.stack.imgur.com/rRQM8.png)
As user202729 pointed out, sometimes these equivalences are somewhat surprising; in particular, the following partitions are equivalent by homeomorphism:
[![An example of two partitions that are homeomorphism.](https://i.stack.imgur.com/DruRe.png)](https://i.stack.imgur.com/DruRe.png)
# Challenge
This [fastest-code](/questions/tagged/fastest-code "show questions tagged 'fastest-code'") challenge will have you write a program that computes as many terms of this new sequence as possible. In case of a close call, I will run submissions on my machine, a 2017 MacBook Pro with 8 GB of RAM.
(Be aware that some of these square partitions [are more complicated](https://en.wikipedia.org/wiki/Squaring_the_square) then just putting two smaller partitions side-by-side.)
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