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%I #33 Jun 30 2020 14:58:38
%S 1,0,7,6,5,3,9,1,9,2,2,6,4,8,4,5,7,6,6,1,5,3,2,3,4,4,5,0,9,0,9,4,7,1,
%T 9,0,5,8,7,9,7,6,5,6,3,2,9,0,1,1,5,0,8,6,6,9,8,5,6,8,1,4,6,9,8,1,9,2,
%U 4,3,4,1,4,6,2,6,4,2,6,4,3,4,1,2,7,7,6,1,9,9,0,4,0,9,1,5,8,7,3,1,9,2,9,6,7
%N Decimal expansion of the one-ninth constant.
%C The generating function of A113184 equals 1/8 at q = Lambda = 0.1076539192... where K(k)=2E(k). - _Michael Somos_, Jul 21 2006
%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 259-262.
%H G. C. Greubel, <a href="/A072558/b072558.txt">Table of n, a(n) for n = 0..10000</a>
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/onenin/onenin.html">The "One-Ninth" Constant</a> [Broken link]
%H Steven R. Finch, <a href="http://web.archive.org/web/20010603070928/http://www.mathsoft.com/asolve/constant/onenin/onenin.html">The "One-Ninth" Constant</a> [From the Wayback machine]
%H Alphonse P. Magnus, Jean Meinguet, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.6610">The elliptic functions and integrals of the '1/9' problem</a>
%H Alphonse P. Magnus, Jean Meinguet, <a href="https://doi.org/10.1023/A:1019141226189">The elliptic functions and integrals of the '1/9' problem</a>, presented at Antwerpen international conference on rational approximation, 1999, ICRA99, Numerical Algorithms 24: (1-2) (2000) 117-139.
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap91.html">The One-ninth constant</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/One-NinthConstant.html">One-Ninth Constant</a>
%e 0.1076539192264845766153234450909471905879...
%t c = k /. FindRoot[ EllipticK[k^2] == 2*EllipticE[k^2], {k, 9/10}, WorkingPrecision -> 120]; Take[ RealDigits[ N[Exp[-Pi*(EllipticK[1 - c^2] / EllipticK[c^2])], 120]][[1]], 105] (* _Jean-François Alcover_, Jul 28 2011, after MathWorld *)
%t RealDigits[q /. FindRoot[4 EllipticE[InverseEllipticNomeQ[q]] == Pi EllipticTheta[3, 0, q]^2, {q, 1/9, 0, 1}, WorkingPrecision -> 105]][[1]] (* _Jan Mangaldan_, Jun 25 2020 *)
%Y Cf. A072559, A073007.
%K cons,nonn
%O 0,3
%A _Robert G. Wilson v_, Aug 03 2002
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