A249137 Decimal expansion of the derivative y'(0) where y(x) is the solution to the differential equation y''(x)+exp(y(x))=0, with y(0)=y(beta)=0 and beta maximum (beta = A249136).
2, 1, 3, 3, 8, 7, 7, 9, 3, 9, 9, 1, 5, 0, 6, 1, 1, 1, 9, 8, 0, 7, 2, 4, 4, 6, 7, 7, 4, 0, 1, 8, 5, 2, 9, 1, 9, 2, 2, 8, 9, 6, 2, 3, 8, 5, 3, 7, 9, 6, 4, 6, 8, 6, 1, 7, 7, 7, 2, 3, 4, 5, 9, 2, 7, 1, 9, 0, 6, 1, 1, 7, 5, 5, 7, 7, 4, 9, 9, 0, 3, 8, 1, 5, 7, 5, 2, 3, 9, 9, 3, 3, 7, 4, 7, 3, 2, 9, 4, 3, 3, 5, 6
Offset: 1
Examples
2.13387793991506111980724467740185291922896238537964686...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..10000
- Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 32.
- Eric Weisstein's MathWorld, Laplace Limit.
Programs
-
Mathematica
digits = 103; lambda = x /. FindRoot[x*Exp[Sqrt[1 + x^2]]/(1 + Sqrt[1 + x^2]) == 1, {x, 1}, WorkingPrecision -> digits+5]; mu = Sqrt[lambda^2 + 1]; RealDigits[Sqrt[2]*Sinh[mu], 10, digits] // First
Formula
y'(0) = sqrt(2)*sinh(sqrt(lambda^2 + 1)), where lambda is A033259, the Laplace limit constant 0.66274...