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

0,1

Links

Table of n, a(n) for n=0..99.

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

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

Example

0.22598715591349733298631152068808233761701168147556791654406413883...

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 *)

Prog

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

Crossrefs

Cf. A222362 (a similar constant).

Sequence in context: A233073 A275266 A223387 * A262614 A184296 A131133

Adjacent sequences: A278383 A278384 A278385 * A278387 A278388 A278389

Keyword

nonn,cons

Author

Jean-François Alcover, Nov 20 2016

Status

approved