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%I #20 Jul 19 2021 08:01:52
%S 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,
%T 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,
%U 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
%N 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).
%C 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.
%C Guide to constants c associated with tangencies:
%C A196610: 1/x and c*cos(x)
%C A196619: 1/x - c and cos(x)
%C A196774: 1/x + c and sin(x)
%C A196625: 1/x and cos(c-x)
%C A196772: 1/x and sin(x+c)
%C A196758: 1/x and c*sin(x)
%C A196765: c/x and sin(x)
%C A196823: 1/(1+x^2) and -c+cos(x)
%C A196914: 1/(1+x^2) and c*cos(x)
%C A196832: 1/(1+x^2) and c*sin(x)
%C A197016: x=0, y=0, and cos(x)
%e c=0.60578021702155370914841756575969877104...
%t Plot[{1/x, Cos[x - 0.60578]}, {x, 0, 2 Pi}]
%t r = GoldenRatio; xt = Sqrt[r];
%t x1 = N[xt, 100]
%t RealDigits[x1] (* A139339 *)
%t c = Sqrt[r] - ArcCsc[r];
%t c1 = N[c, 100]
%t RealDigits[c1] (* A196625 *)
%t slope = N[r - Sqrt[5], 100]
%t RealDigits[slope] (* -1+A001622; -1+golden ratio *)
%Y Cf. A139339, A196772.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Oct 05 2011
%E a(99) corrected by _Georg Fischer_, Jul 19 2021
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