%I
%S 2,1,1,2,2,4,6,8,6,8,14,25,36,180,76,80,66,2068,354,7316
%N Number of cycles for the map LL:x>x^22 acting on Z/(2^n1).
%C A cycle is the orbit of an element x of Z/(2^n1) such x=LL^c(x) for some positive integer c, i.e. { x, LL(x), ..., LL^c(x)=x }.
%H Troy Vasiga and Jeffrey Shallit: <a href="http://www.cs.uwaterloo.ca/~tmjvasig/papers/newvasiga.pdf">On the iteration of certain quadratic maps over GF(p)</a>, Discrete Math. (277) 219240.
%F If p=2^n1 is prime, then a(n) = 1/2 + sum_{d2^(n1)1} eulerphi(d)/ordp(2,d)/2, where ordp(2,d) = min { e in N*  2^e=1 (mod d) or 2^e=1 (mod d) }
%e a(0)=2 since fixed points 2 and 1 are the only cycles for LL on Z/(0) = Z;
%e a(1)=1 since Z/(1) = {0};
%e a(2)=1 since 2=1 is a cycle of length 1 (fixed point) for LL on Z/(3) and LL(0)=2=1, LL(1)=1;
%e a(3)=2 since 3,4(=3) > 0 > 5(=2) > {2} and 1 > {6(=1)} for LL acting on Z/(7);
%e a(5)=4 since {2}, {30}, {12,18} and {3,7,16,6} are the cycles for LL acting on Z/(31).
%o (PARI) numcycles(q) = { local(Mq=2^q1, v=vector(Mq+1), c=1, i, start, cyc=0); if(q<2,return(1+!q)); for( j=1, #v, if(v[j],next); i=j; start=c; until(v[i=1+((i1)^22)%Mq],v[i]=c++); if(v[i]>start, cyc++)); cyc } A128976=vector(20,i,numcycles(i1))
%Y Cf. A003010.
%K more,nonn
%O 0,1
%A _M. F. Hasler_, Apr 29 2007, corrected May 19 2007
