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).

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, 2, 7, 8, 7

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..102.

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 * A343948 A195368 A198426

Adjacent sequences: A196622 A196623 A196624 * A196626 A196627 A196628

Keyword

nonn,cons

Author

Clark Kimberling, Oct 05 2011

Extensions

a(99) corrected by Georg Fischer, Jul 19 2021

Status

approved