A248618 -id:A248618 - 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

A258428 Decimal expansion of arctan(1/e).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 31 2015

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

			0.35251342177761899747085992272395350035945275021939640543781203320...
In degrees:
20.1975312895726498111743461325943940876066625590453644638010046029...

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

Cf. A001113, A257777, A248618.

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

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

Equals Integral_{x=1..oo} 1/(2*cosh(x)) dx. - Amiram Eldar, Aug 10 2020

A248617 Decimal expansion of the solution when Gudermannian(x) equals 1.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Oct 09 2014

Inverse of A248618.

			1.22619117088351707081306096747190675272424835022074027913861684354298467624428...

Wikipedia, Gudermannian function.

Cf. A248618.

evalf(log(tan((2+Pi)/4)),100) # Vaclav Kotesovec, Oct 11 2014

RealDigits[ InverseGudermannian[ 1], 10, 111][[1]]

Equals log(tan((2+Pi)/4)). - Vaclav Kotesovec, Oct 11 2014
From Amiram Eldar, Apr 07 2022: (Start)
Equals 2*arctanh(tan(1/2)).
Equals Integral_{x=0..1} sec(x) dx. (End)

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A257777 Decimal expansion of arctan(e).

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A258428 Decimal expansion of arctan(1/e).

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A248617 Decimal expansion of the solution when Gudermannian(x) equals 1.

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