OFFSET
1,3
COMMENTS
Equals the inverse of the position xm of the Dawson integral maximum, and also the negative of the second derivative of the Dawson integral at xm. - Stanislav Sykora, Sep 17 2014
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.9 Hyperbolic volume constants, p. 512.
LINKS
Stanislav Sykora, Table of n, a(n) for n = 1..2000
Eric Weisstein's MathWorld, Dawson's Integral
Wikipedia, Dawson function
FORMULA
Volume(n) ~ 2*sqrt(Pi)*c^n/GAMMA((n+1)/2), where GAMMA is the Euler gamma function.
EXAMPLE
1.0820884492703633969455186604829543727812...
MATHEMATICA
digits = 100; DawsonF[x_] := Sqrt[Pi]*Erfi[x]/(2*Exp[x^2]); c = 2*DawsonF[x] /. FindRoot[DawsonF'[x], {x, 1}, WorkingPrecision -> digits+5]; RealDigits[c, 10, digits] // First
PROG
(PARI) Erfi(z) = -I*(1.0-erfc(I*z));
Dawson(z) = 0.5*sqrt(Pi)*exp(-z*z)*Erfi(z);
DDawson(z) = 1.0 - 2*z*Dawson(z); \\ Derivative of the above
x = 1.0/solve(z=0.1, 2.0, real(DDawson(z))) \\ Stanislav Sykora, Sep 17 2014
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Jun 05 2014
STATUS
approved