A196915 -id:A196915 - cp's OEIS frontend

A196914 Decimal expansion of the number c for which the curve y=1/(1+x^2) is tangent to the curve y=ccos(x), and 0 < x < 2Pi.

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

Offset: 0

Clark Kimberling, Oct 07 2011

			c=0.87446003662400926554814644840673725663073...

Cf. A196913, A196915.

Plot[{1/(1 + x^2), 0.874*Cos[x]}, {x, .5, 1}]
t = x /. FindRoot[Tan[x] == 2 x/(1 + x^2), {x, .5, 1}, WorkingPrecision -> 100]
RealDigits[t]    (* A196913 *)
c = N[Sqrt[t^4 + 6 t^2 + 1]/(t^4 + 2 t^2 + 1), 100]
RealDigits[c]    (* A196914 *)
slope = N[-c*Sin[t], 100]
RealDigits[slope](* A196915 *)

x=0.87446003662400926554814644840673725663073...

A196913 Decimal expansion of the number x satisfying 0 < x < 2Pi and 2x = (1 + x^2)tan(x).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 07 2011

			x=0.7682171553153782504312122866979254095466915658...

Cf. A196816, A196914.

Plot[{1/(1 + x^2), 0.874*Cos[x]}, {x, .5, 1}]
t = x /. FindRoot[Tan[x] == 2 x/(1 + x^2), {x, .5, 1}, WorkingPrecision -> 100]
RealDigits[t]    (* A196913 *)
c = N[Sqrt[t^4 + 6 t^2 + 1]/(t^4 + 2 t^2 + 1), 100]
RealDigits[c]    (* A196914 *)
slope = N[-c*Sin[t], 100]
RealDigits[slope](* A196915 *)

cp's OEIS Frontend

A196914 Decimal expansion of the number c for which the curve y=1/(1+x^2) is tangent to the curve y=ccos(x), and 0 < x < 2Pi.

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A196913 Decimal expansion of the number x satisfying 0 < x < 2Pi and 2x = (1 + x^2)tan(x).

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cp's OEIS Frontend

A196914 Decimal expansion of the number c for which the curve y=1/(1+x^2) is tangent to the curve y=c*cos(x), and 0 < x < 2*Pi.

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A196913 Decimal expansion of the number x satisfying 0 < x < 2*Pi and 2x = (1 + x^2)*tan(x).

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A196914 Decimal expansion of the number c for which the curve y=1/(1+x^2) is tangent to the curve y=ccos(x), and 0 < x < 2Pi.

A196913 Decimal expansion of the number x satisfying 0 < x < 2Pi and 2x = (1 + x^2)tan(x).