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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 short list of problems I've been interested in, with graphics included if possible.

A261752 (Knight's Domination): "Minimum number of knights on an n X n chessboard such that every square is attacked."

A004670 (and friends): "Theta series of Barnes-Wall lattices in dimension 2^n."

Dimension (OEIS)

16 (A008409)

32 (A004670)

64 (A103936)

Other OEIS Problems I Find Interesting But Don't Have Any Graphics For