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User: Gordon Roesler

Gordon Roesler has authored 2 sequences.

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

Gordon Roesler, Jul 23 2012

nonn

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

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

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

More terms from Michel Marcus, Aug 29 2013

A213082 Values of n for which the number of roots of the function sin(x)/x - 1/n increases.

Original entry on oeis.org

1, 2, 8, 15, 21, 27, 33, 40, 46, 52, 59, 65, 71, 77, 84, 90, 96, 103, 109, 115, 121, 128, 134, 140, 147, 153, 159, 165, 172, 178, 184, 191, 197, 203, 209, 216, 222, 228, 235, 241, 247, 253, 260, 266, 272, 279, 285, 291, 297, 304, 310, 316, 323, 329, 335

Offset: 1

Gordon Roesler, Jul 22 2012

nonn

Maxima of sin(x)/x correspond to odd solutions of x(m)=tan(x(m)). At maxima, sin(x(m))/x(m)=sin(tan(x(m)))/tan(x(m)). Number of roots of f(x)=sin(x)/x - 1/n increases when n = int(x(m)/sin(x(m))+1.

			For n=1 there is 1 root, for n=2,...7 there are 2 roots, for n=8,...14 there are 6 roots, etc.

t = Table[x*Cos[x] - Sin[x], {x, 400}]; t2 = {1, 2}; Do[If[t[[n]] > 0 && t[[n + 1]] < 0, AppendTo[t2, n + 1]], {n, Length[t] - 1}]; t2 (* T. D. Noe, Jul 30 2012 *)

Terms after a(8) from T. D. Noe, Jul 30 2012

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User: Gordon Roesler

A214628 Intersections of radii with the cycloid.

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