A214628 Intersections of radii with the cycloid.

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, 18, 18, 20, 20, 20, 22, 22, 22, 22, 24, 24, 24, 26, 26, 26, 28, 28, 28, 30, 30, 30, 32, 32, 32, 34, 34, 34, 36, 36, 36, 36, 38, 38, 38, 40, 40, 40

Offset: 1

Author

Gordon Roesler, Jul 23 2012

Keywords

• nonn

Comments

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).

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

Examples

			For n=1..4, a(n)=2; for n=5..7, a(n)=4.
		

Programs

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));}
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

Extensions

More terms from Michel Marcus, Aug 29 2013