%I #13 Jun 25 2021 23:15:39
%S 2,2,2,2,4,4,4,6,6,6,8,8,8,8,10,10,10,12,12,12,14,14,14,16,16,16,18,
%T 18,18,20,20,20,22,22,22,22,24,24,24,26,26,26,28,28,28,30,30,30,32,32,
%U 32,34,34,34,36,36,36,36,38,38,38,40,40,40
%N Intersections of radii with the cycloid.
%C Number of times the line y=x/n intersects the cycloid specified by x=tsin(t), y=1cos(t) or, by symmetry, number of times the line y=n*x intersects the cycloid specified by x=1cos(t), y=tsin(t). It is equal to twice the number of arches that are intersected by the lines (2 intersection points by arch).
%C To find this sequence one can look for the slopes of the tangents to the nth arch when these tangents pass through the origin (see PARI script). If one consider the indices where a(n) change value, one gets: 1, 4, 7, 10, 14, 17, 20, 23, 26, ... that may well be A062389, as this is the slope of the line joining the origin to the summit of the nth arch. Will this be true for all n?  _Michel Marcus_, Aug 29 2013
%e For n=1..4, a(n)=2; for n=5..7, a(n)=4.
%o (PARI) slop(n) = {ang = 2*n*Pi; val = solve(x=ang + Pi/100, ang + Pi, 2  2*cos(x)  x*sin(x)); vinvn = floor((1  cos(val))/sin(val));}
%o lista(nn) = {nbc = 0; nbi = 1; for (i=1, nn, nnbc = slop(i); for (j = 1, nnbc  nbc, print1(2*nbi, ", ")); nbi++; nbc = nnbc;);} \\ _Michel Marcus_, Aug 29 2013
%K nonn
%O 1,1
%A _Gordon Roesler_, Jul 23 2012
%E More terms from _Michel Marcus_, Aug 29 2013
