A278386 Decimal expansion of the excess of the exponential curve arc length over the length of the x-axis from -infinity to zero.

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

Offset: 0

Author

Jean-François Alcover, Nov 20 2016

Keywords

• nonn
• cons

Examples

			0.22598715591349733298631152068808233761701168147556791654406413883...
		

Links

• Jean-François Alcover, Involute of the exponential curve (left branch).

Crossrefs

Cf. A222362 (a similar constant).

Programs

Mathematica

RealDigits[Sqrt[2] - 1 + Log[2] - Log[1 + Sqrt[2]], 10, 100][[1]]
RealDigits[Sqrt[2] - 1 - ArcSinh[7/(4 (5 + 3 Sqrt[2]))], 10, 100][[1]] (* Jan Mangaldan, Nov 22 2020 *)

PARI

sqrt(2) - 1 + log(2) - log(1 + sqrt(2)) \\ Michel Marcus, Nov 20 2016

Formula

Equals Integral_{-infinity..0} (sqrt(1 + exp(2x))-1) dx.

Equals sqrt(2) - 1 + log(2) - log(1 + sqrt(2)).

Equals sqrt(2) - 1 - arcsinh(7/(4*(5 + 3*sqrt(2)))). - Jan Mangaldan, Nov 23 2020

Equals sqrt(2) - 1 - arcsinh((5 - 3*sqrt(2))/4). - Vaclav Kotesovec, Nov 27 2020

Equals Integral_{x=0..1} (sqrt(x^2 + 1) - 1)/x dx. - Kritsada Moomuang, May 27 2025