cp's OEIS Frontend

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

%I A278386 #32 May 28 2025 04:31:34
%S A278386 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,
%T A278386 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,
%U A278386 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
%N A278386 Decimal expansion of the excess of the exponential curve arc length over the length of the x-axis from -infinity to zero.
%H A278386 Jean-François Alcover, <a href="/A278386/a278386.pdf">Involute of the exponential curve</a> (left branch).
%F A278386 Equals Integral_{-infinity..0} (sqrt(1 + exp(2x))-1) dx.
%F A278386 Equals sqrt(2) - 1 + log(2) - log(1 + sqrt(2)).
%F A278386 Equals sqrt(2) - 1 - arcsinh(7/(4*(5 + 3*sqrt(2)))). - _Jan Mangaldan_, Nov 23 2020
%F A278386 Equals sqrt(2) - 1 - arcsinh((5 - 3*sqrt(2))/4). - _Vaclav Kotesovec_, Nov 27 2020
%F A278386 Equals Integral_{x=0..1} (sqrt(x^2 + 1) - 1)/x dx. - _Kritsada Moomuang_, May 27 2025
%e A278386 0.22598715591349733298631152068808233761701168147556791654406413883...
%t A278386 RealDigits[Sqrt[2] - 1 + Log[2] - Log[1 + Sqrt[2]], 10, 100][[1]]
%t A278386 RealDigits[Sqrt[2] - 1 - ArcSinh[7/(4 (5 + 3 Sqrt[2]))], 10, 100][[1]] (* _Jan Mangaldan_, Nov 22 2020 *)
%o A278386 (PARI) sqrt(2) - 1 + log(2) - log(1 + sqrt(2)) \\ _Michel Marcus_, Nov 20 2016
%Y A278386 Cf. A222362 (a similar constant).
%K A278386 nonn,cons
%O A278386 0,1
%A A278386 _Jean-François Alcover_, Nov 20 2016