A389268 Decimal expansion of x > 0 satisfying 2*x^2 + 1 = exp(x^2).

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

Offset

1,3

Comments

Also the decimal expansion of x that maximizes f(x) = (1-exp(-x^2))/x.

The value of x is the coefficient in equation (8) of McEachran et al. (2020), and x*sqrt(3)/2 is the coefficient in equation (6) of the same paper. This problem arises from maximizing the effectiveness of nitrate removal in a saturated riparian buffer (SRB) whose normalized length is x; the effectiveness involves the flow through the SRB, which by Darcy's law is proportional to 1/x, and the concentration leaving the SRB, which is proportional to exp(-x^2).

References

A. R. McEachran, L. C. Dickey, C. R. Rehmann, T. A. Groh, T. M. Isenhart, M. A. Perez, and C. J. Rutherford, Improving the effectiveness of saturated riparian buffers for removing nitrate from subsurface drainage, Journal of Environmental Quality, 49 (2020), 1624-1632.

Links

Chris R. Rehmann, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Lambert W-Function.

Formula

Equals sqrt(A202343).

Example

1.1209064227785340319767667356906333599337977824722...

Mathematica

RealDigits[Sqrt[-1/2 - ProductLog[-1, -1/(2*Sqrt[E])]], 10, 99] // First

Prog

(MATLAB) fzero(@(x) 1+2*x.^2-exp(x.^2), 1)
(PARI) solve(x=1, 2, 2*x^2+1-exp(x^2))

Crossrefs

Cf. A202320, A202343.

Sequence in context: A140415 A105819 A153616 * A190258 A161119 A019750

Adjacent sequences: A389265 A389266 A389267 * A389269 A389270 A389271

Keyword

nonn,cons

Author

Chris R. Rehmann, Sep 27 2025

Status

approved