I often use the Fibonacci numbers mod 10 when I need to do this in my head. It isn't great -- odd numbers are twice as frequent (because mod 2 they're [1,1,0])

I often use the Fibonacci numbers mod 10 when I need to do this in my head. It isn't great -- odd numbers are twice as frequent (because mod 2 they're [1,1,0]) and you get sequences like [7,0,7,7], but easy to keep track of and the cycle is 60 elements long. The other one I use sometimes is the powers of 2 mod 100 and use the leading digit. This gives a cycle length of 20 with a flat distribution.