Pentagonal Number Theorem
Euler's pentagonal number theorem is an amazing recursion formula. The same recursion also applies for the divisor function. In the video, I comment, how I discovered the formula using Cuisenaire material in high school. Pictures shown at the beginning and end were taken early in the morning in Cambridge (during sun rise).
P..S only noticed after uploading that the short reformulation of the pentagonal number theorem should be sum_k(-1)^k p(n-f(k)) = 0 (middle box, as this means p(n)=p(n-f(1))+p(n-f(-1))-p(n-f(2))-p(n-f(3)) etc - It is a super sum, not a sum. By the way, since the partition function is a counting function, like Euler characteristic, it begs the comparison of zero super sums for Euler characteristics, like for odd dimensional manifolds. Associations to to particle physics are not completely out of bound as p(n) counts the number of irreducible representations of the symmetric group S(n) and we know from Wigner that irreducible representations of groups can be seen as "particles".