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%I #35 Mar 21 2022 11:48:49
%S 1,5,8,8,0,7,1,0,2,2,6,1,1,3,7,5,3,1,2,7,1,8,6,8,4,5,0,9,4,2,3,9,5,0,
%T 1,0,9,4,5,2,7,4,6,6,2,1,6,7,4,8,2,5,6,1,6,7,6,5,6,7,7,4,1,8,1,6,6,5,
%U 5,1,9,6,1,1,5,4,3,0,9,2,6,2,3,3,2,0,3
%N Negative initial slope of the Thomas-Fermi equation, y"(x) = Sqrt( y(x)^3 / x), with boundary conditions y(0) = 1 and y(Infinity) = 0.
%D C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer, 1999, p. 167.
%D Max Born, Atomic Physics, Blackie & Son Ltd., 8th. ed., 1969, p. 200.
%D Hagen Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th edition, World Scientific (Singapore, 2009), p. 422.
%D J. Schwinger, Quantum Mechanics: Symbolism of Atomic Measurements, Springer (2001), p. 419.
%H Peter J. C. Moses, <a href="/A231786/b231786.txt">Table of n, a(n) for n = 1..5000</a>
%H P. Amore, J. Boyd, and F. Fernández, <a href="http://arxiv.org/abs/1205.1704">Accurate calculation of the solutions to the Thomas-Fermi equations</a>, arXiv:1205.1704 [quant-ph], 2012-2014.
%H John P. Boyd, <a href="https://doi.org/10.1016/j.cam.2012.11.015">Rational Chebyshev series for the Thomas-Fermi function: Endpoint singularities and spectral methods</a>, Journal of Computational and Applied Mathematics, 244 (2013), p. 90-101.
%H S. Esposito, <a href="https://doi.org/10.1119/1.1484144">Majorana solution of the Thomas-Fermi equation</a>, Am. J. Phys. 70 (8), p 852-856. (2002); arXiv:<a href="https://arxiv.org/abs/physics/0111167">physics/0111167</a> [physics.atom-ph], 2001.
%H R. P. Feynman, N. Metropolis, and E. Teller, <a href="http://authors.library.caltech.edu/3519/1/FEYpr49a.pdf">Equations of State of Elements Based on the Generalized Fermi-Thomas Theory</a>, Phys. Rev. 75, p 1561-1573 (1949); doi:<a href="https://doi.org/10.1103/PhysRev.75.1561">10.1103/PhysRev.75.1561</a>.
%H J. Schwinger, <a href="https://doi.org/10.1103/PhysRevA.22.1827">Thomas-Fermi model: The Leading correction</a>, Phys. Rev A, vol 22 (1980), p. 1827-1832.
%e y'(0) = -1.588...
%t nn = 150; Clear[a]; a[0] = 1; a[1] = 9 - Sqrt[73]; (* a[m]:=N[_,x] will give about x/3 digits and will calculate up to about a[4 x]. Example: to find 1000 digits, x needs to be 3000 and will calculate a[m] upto about a[12500] *) a[m_] := a[m] = N[(Sum[((n + 7) a[n - 1] + (n + 1) a[n + 1] - 2 (n + 4) a[n]) a[m - n], {n, m - 2}] + (a[1] (m + 6)) a[m - 2] + ((m + 7) - 2 a[1] (m + 3)) a[m - 1])/(2 (m + 8) - a[1] (m + 1)), nn]; RealDigits[N[(3/16)^(1/3) Sum[a[n], {n, 0, #[[2]]}], #[[1]]] &[{-MantissaExponent[a[#]][[2]] - 1, #} &[NestWhile[# + 1 &, 0, Precision[a[#]] > 5 &] - 1]]][[1]]
%K nonn,cons
%O 1,2
%A _Peter J. C. Moses_, Nov 13 2013
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