A257776 -id:A257776 - cp's OEIS frontend

A019798 Decimal expansion of sqrt(2*e).

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

Offset: 1

N. J. A. Sloane

The coefficient a for which y=a*sqrt(x) kisses the exponential function y=exp(x). The kissing point is (0.5, sqrt(e)). For more details, see A257776. Also, inverse of this constant equals the maximum value of sqrt(x)*exp(-x) for positive x, attained at x=1/2. - Stanislav Sykora, Nov 04 2015

			2.3316439815971242033635360621684008763802362991875884230...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers.

Cf. A001113, A019774, A257775, A257776.

SetDefaultRealField(RealField(100)); R:= RealField(); Sqrt(2*Exp(1)); // G. C. Greubel, Sep 08 2018

RealDigits[Sqrt[2*E], 10, 100][[1]] (* G. C. Greubel, Sep 08 2018 *)

sqrt(2*exp(1)) \\ Michel Marcus, Nov 05 2015

From Amiram Eldar, Jul 08 2023: (Start)
Equals Product_{n>=0} (e / (1 + 1/(n-1/2))^n).
Equals Product_{n>=0} (e * (1 - 1/(n+1/2))^n). (End)

A257777 Decimal expansion of arctan(e).

Original entry on oeis.org

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

A257775 Decimal expansion of (e/2)^2.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, May 12 2015

The coefficient a of the unique parabola y = a*x^2 which, at some x > 0, kisses the exponential function y = exp(x). The kissing point coordinates are (2,e^2).

			1.847264024732662556807606865143751953295078892637961831021781955630...

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Cf. A001113, A019739, A257776, A257777.

RealDigits[Exp[2]/4, 10, 120][[1]] (* Amiram Eldar, Jun 26 2023 *)

PARI
```
exp(2)/4
```

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A019798 Decimal expansion of sqrt(2*e).

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A257777 Decimal expansion of arctan(e).

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A257775 Decimal expansion of (e/2)^2.

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