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%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=t-sin(t), y=1-cos(t) or, by symmetry, number of times the line y=n*x intersects the cycloid specified by x=1-cos(t), y=t-sin(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 n-th 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 n-th 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
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