A258428 -id:A258428 - cp's OEIS frontend

A257777 Decimal expansion of arctan(e).

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

Offset: 1

Stanislav Sykora, May 12 2015

The slope of the unique straight line passing through the origin which kisses the exponential function y=exp(x), i.e., the angle (in radians) the tangent line subtends with the X axis. The kissing point coordinates are (1,e).

			1.21828290501727762176046176891579794173913194946815650504966...
In degrees:
69.8024687104273501888256538674056059123933374409546355361989953970...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Robert Frontczak, Further results on arctangent sums with applications to generalized Fibonacci numbers, Notes on Number Theory and Discrete Mathematics, Vol. 23, No. 1 (2017), pp. 39-53.

Cf. A001113, A248618, A257775, A257776, A257896, A258428.

RealDigits[ArcTan[E],10,105][[1]] (* Vaclav Kotesovec, Jun 02 2015 *)

PARI
```
atan(exp(1))
```

Equals (Sum_{k>=0} arctan(sinh(1)/cosh(k))) - Pi/4 (Frontczak, 2017, eq. (3.22)). - Amiram Eldar, Jul 09 2023

A248618 Decimal expansion of the solution when inverse Gudermannian(x) equals 1.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Oct 09 2014

Inverse of A248617.

This is the angle of the unique wedge having its apex at the origin and kissing the exponential curve y=exp(x) on one side, and its inverse logarithmic function y=log(x) on the other side. - Stanislav Sykora, May 31 2015

			0.86576948323965862428960184619184444137967919924876009961184822974244822945841...
The wedge angle in degrees:
49.6049374208547003776513077348112118247866748819092710723979907940346891648208... - _Stanislav Sykora_, May 31 2015

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Wikipedia, Gudermannian function.

Cf. A248617, A257777, A258428.

evalf(arcsin((exp(2)-1)/(exp(2)+1)),100) # Vaclav Kotesovec, Oct 11 2014

RealDigits[ Gudermannian[ 1], 10, 111][[1]]

asin((exp(2)-1)/(exp(2)+1)) \\ Michel Marcus, Oct 11 2014

atan(exp(1))-atan(1/exp(1)) \\ Stanislav Sykora, May 31 2015

2*atan(exp(1))-Pi/2 \\ Charles R Greathouse IV, Jun 02 2015

Equals arcsin((exp(2)-1)/(exp(2)+1)). - Vaclav Kotesovec, Oct 11 2014
Equals atan(e)-atan(1/e) = A257777-A258428. - Stanislav Sykora, May 31 2015
From Amiram Eldar, Apr 07 2022: (Start)
Equals 2*arctan(tanh(1/2)).
Equals Integral_{x=0..1} sech(x) dx. (End)

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A257777 Decimal expansion of arctan(e).

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A248618 Decimal expansion of the solution when inverse Gudermannian(x) equals 1.

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