Fibonaci implemented in brainfuck and formalized in isabelle and some other stuff
Very simple one, modulo 256
+>>+<< <<++++++[->>
[<+>>+<-] < [->+<] > >> [->+<] > [-<<<+>>>] << [->+<] <.
<<]
"Fields" used: [_C_|_a'_|_a_|_a"_|_b_|_b'_]
, where x' is a copy of x, and C the number of iterations