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Does $1^2+2^2+3^2+\cdots+24^2=70^2$ or a simple equivalent have an interesting categorification?

Lots of combinatorical identities do. For instance, the Vandermonde convolution identity and the binomial theorem can both be reinterpreted as isomorphisms of $S_m\times S_n$-sets. Obviously this identity involves particular numbers, so conceivably if it did have a categorification it would be some kind of "exceptional" situation that doesn't work for arbitrarily-sized sets. And also maybe instead of using a full symmetric group we can use a "big enough" group. (For instance, I guess there's a Mathieu group that acts very transitively on $24$ points, perhaps that can be related to $(2n)(2n+1)(2n+2)=24\cdot70^2$ for $n=24$.)

I'll use the (soft-question) tag since this seems pretty subjective.

anon
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    As a simple observation, we have that $k^2=2\binom{k}{2}+k=2\binom{k}{2}+\binom{k}{1}.$ So your sum can be seen as $$\sum _{k=1}^n\left (2\binom{k}{2}+\binom{k}{1}\right )=2\binom{n+1}{3}+\binom{n+1}{2}=\binom{n+1}{3}+\binom{n+2}{3}$$ So you are asking why the last number is a square? – Phicar Dec 08 '20 at 23:21
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    That's a nice observation. In that case, I guess I'd roughly be asking for an isomorphism of $G$-sets between $\binom{24+1}{3}+\binom{24+2}{3}$ and $70^2$, where we interpret $G$ as a big subgroup of $S_{24}$ which also somehow acts on a $70$-element set. – anon Dec 08 '20 at 23:25
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    A search has turned up this: https://en.m.wikipedia.org/wiki/Cannonball_problem ., with a relation to https://en.m.wikipedia.org/wiki/Leech_lattice . –  Dec 09 '20 at 00:08
  • You may also like this question with a lot of wonderful answers: https://math.stackexchange.com/questions/505367/surprising-identities-equations/505421#505421 – imranfat Dec 09 '20 at 02:21

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