Programming exercise - Fixed points

The function F is defined by

F(1) = 1
F(3) = 3
F(2*n) = F(n)
F(4*n+1) = 2*F(2*n+1)-F(n)
F(4*n+3) = 3*F(2*n+1)-2*F(n)

A fixed point is n such as F(n) = n
How many fixed points less or equal to 10^100 have F?

Answer format: the count

[My timing: < 100 ms]