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A196625 Decimal expansion of the number c for which the curve y=1/x is tangent to the curve y=cos(x-c), 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 (list; constant; graph; refs; listen; history; text; internal format)
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(x-c) at (u,v), and the slope of the tangent line is r-1.

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(c-x)

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

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Last modified June 26 23:21 EDT 2017. Contains 288777 sequences.