The Online Encyclopedia of Integer Sequences is an online database of integer-valued sequences that anyone can contribute to. It's got a lot of interesting problem statements, many with only partial solutions. I've compiled a list of some problems I've been interested in, with graphics included if possible.
A261752 (Knight's Domination)
"Minimum number of knights on an $n\times n$ chessboard such that every square is attacked."
A004670 (and friends): Barnes-Wall Lattice Theta Series
Theta series of Barnes-Wall lattices in dimension $n=2^k$. A theta series of a lattice counts the number of points with a given squared distance from the origin. For example, in the typical square grid lattice $\mathbb{Z}^2$ there is one point of square distance 0, 4 points of square distance 1, 4 points of square distance 2, etc. This is encoded in the function $1+4z+4z^2+4z^4+8z^5+\ldots$. Theta functions have some nice mathematical properties which ultimately arise from the symmetries of the lattices they arise from. The Barnes-Wall lattices are a special class of lattices of dimension $2^k$. Since they are even lattices, they never have vectors of odd square length, so I adopt the convention to list their theta functions in powers of $q = z^2$. These plots were generated with Sage.
OEIS Sequences I've Worked On Recently
Many of these didn't amount to contributions, but I was at least (usually) able to verify the listed terms.
Other OEIS Sequences Near and Dear to My Heart
These are sequences related to my undergrad thesis.