

A196625


Decimal expansion of the number c for which the curve y=1/x is tangent to the curve y=cos(xc), and 0<x<2*pi; c=sqrt(r)arccsc(r), where r=(1+sqrt(5))/2 (the golden ratio).


2



6, 0, 5, 7, 8, 0, 2, 1, 7, 0, 2, 1, 5, 5, 3, 7, 0, 9, 1, 4, 8, 4, 1, 7, 5, 6, 5, 7, 5, 9, 6, 9, 8, 7, 7, 1, 0, 4, 8, 1, 1, 7, 9, 0, 3, 1, 1, 4, 1, 4, 8, 4, 0, 5, 7, 8, 5, 1, 6, 6, 5, 3, 9, 7, 3, 5, 3, 1, 8, 5, 8, 6, 1, 5, 7, 0, 0, 8, 7, 3, 0, 1, 2, 2, 4, 7, 7, 3, 8, 3, 8, 1, 8, 8, 7, 9, 1, 2, 3, 3
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OFFSET

0,1


COMMENTS

Let r=(1+sqrt(5))/2, the golden ratio. Let u=sqrt(r) and v=1/x. Let c=sqrt(r)arccsc(r). The curve y=1/x is tangent to the curve y=cos(xc) at (u,v), and the slope of the tangent line is r1.
Guide to constants c associated with tangencies:
A196610: 1/x and c*cos(x)
A196619: 1/x  c and cos(x)
A196774: 1/x + c and sin(x)
A196625: 1/x and cos(cx)
A196772: 1/x and sin(x+c)
A196758: 1/x and c*sin(x)
A196765: c/x and sin(x)
A196823: 1/(1+x^2) and c+cos(x)
A196914: 1/(1+x^2) and c*cos(x)
A196832: 1/(1+x^2) and c*sin(x)
A197016: x=0, y=0, and cos(x)


LINKS

Table of n, a(n) for n=0..99.


EXAMPLE

c=0.60578021702155370914841756575969877104...


MATHEMATICA

Plot[{1/x, Cos[x  0.60578]}, {x, 0, 2 Pi}]
r = GoldenRatio; xt = Sqrt[r];
x1 = N[xt, 100]
RealDigits[x1] (* A139339 *)
c = Sqrt[r]  ArcCsc[r];
c1 = N[c, 100]
RealDigits[c1] (* A196625 *)
slope = N[r  Sqrt[5], 100]
RealDigits[slope] (* 1+A001622; 1+golden ratio *)


CROSSREFS

Cf. A139339, A196772.
Sequence in context: A153754 A096410 A098468 * A195368 A198426 A019110
Adjacent sequences: A196622 A196623 A196624 * A196626 A196627 A196628


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, Oct 05 2011


STATUS

approved



