The OEIS has a lot of cool sequences. Here are some examples I dabbled with:
■ Some sequences which hide Sierpińskish fractals:
 A327853 (Sierpiński's triangle × triangular numbers)
 A327889 (Pascal's triangle ° digit transformation)

■ Noteworthy rare primes (conjectured to be infinite):
 A000000 (WallSunSun primes  none are known)

■ Consecutive palindromes problem:
 A279092 (k>=2, the double palindromes)
 A279093 (k>=3, the triple palindromes)
 A323742 (k=3, the existence problem)
 A333512 (k=3, but no other palindromic bases)
 A327810 (irregular consecutive palindromes)

■ Number formation with basic operations (+,,*,/) problem:
 A142153 (use integers at most once)
 A309885 (use integers exactly once)
 A309886 (use powers of two exactly once)

■ Memorable "Jumping" sequences:

■ Counting on Circles:
 A250001 (arrangements in affine plane)
 A067310 (chords with k simple intersections)
 A331702 (intersections on polygons)

■ Counting with Dominoes:
 A004003 (count domino tilings of 2n X 2n square)
 A333837 (count ways to collapse a triangular formation)

■ Sums of (unordered) factorizations:
 A337037 (numbers with all unique sums)
 A337080 (numbers with some duplicate sums)
 A337081 (primitive complement of such numbers)

■ Puzzlerelated sequences:
