A072558 Decimal expansion of the one-ninth constant.
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, 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, 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
Offset: 0
Examples
0.1076539192264845766153234450909471905879...
References
- Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 259-262.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..10000
- Steven R. Finch, The "One-Ninth" Constant [Broken link]
- Steven R. Finch, The "One-Ninth" Constant [From the Wayback machine]
- Alphonse P. Magnus, Jean Meinguet, The elliptic functions and integrals of the '1/9' problem
- Alphonse P. Magnus, Jean Meinguet, The elliptic functions and integrals of the '1/9' problem, presented at Antwerpen international conference on rational approximation, 1999, ICRA99, Numerical Algorithms 24: (1-2) (2000) 117-139.
- Simon Plouffe, The One-ninth constant
- Eric Weisstein's World of Mathematics, One-Ninth Constant
Programs
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Mathematica
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 *) 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 *) -
PARI
c=solve(x=.9,.91, ellK(x)-2*ellE(x)); exp(-Pi*ellK(sqrt(1 - c^2))/ellK(c)) \\ Charles R Greathouse IV, Feb 04 2025
Comments