Exact iterates [in iterated maps] For any integer a the n th iterate of x  FractionalPart[a x] can be written as FractionalPart[a n x] , or equivalently 1/2 - ArcTan[Cot[a n π x]]/ π .

The patterns are exactly repetitive only when Tan[ θ ]  u/v , where u and v are elements of a primitive Pythagorean triple (so that u , v and Sqrt[u 2 + v 2 ] are all integers, and GCD[u, v]  1 ).

For rational functions f[x] , Integrate[f[x], {x, 0, 1}] must always be a linear function of Log and ArcTan applied to algebraic numbers ( f[x] = 1/(1 + x 2 ) for example yields π /4 ).

The continued fractions for Exp[2/k] and Tan[1/2k] have simple forms (as discussed by Leonhard Euler in the mid-1700s); other rational powers of E and tangents do not appear to.