cp's OEIS Frontend

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).

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