restart: n := 5: # How many terms of the orbits.
 # eight_forward is the forward orbit of g^(k)(8);
 # eight_backward the backward orbit.





 g := proc(n:posint)
 local m;
 m := modp(n,3);
 if m = 0 then 2*n/3;
 elif m=1 then (4*n-1)/3;
 else (4*n+1)/3;
 fi;
 end:
 
 
 h := proc(n:posint)
 local m;
 m := modp(n,4);
 if m = 1 then (3*n+1)/4;
 elif m=3 then (3*n-1)/4;
 else 3*n/2;
 fi;
 end:
 
 eight_forward := proc(n:posint)
 local m,r; r := [8];
 for m from 1 to n do 
 r := [op(r),g(r[nops(r)])];
 od;
 r;
 end:
 
 
 eight_backward := proc(n:posint)
 local m,r; r := [8];
 for m from 1 to n do 
 r := [op(r),h(r[nops(r)])];
 od;
 r;
 end:
 
 ``(eight_forward)(n) = eight_forward(n);

 ``(eight_backward)(n) = eight_backward(n);