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

0,3

Comments

The generating function of A113184 equals 1/8 at q = Lambda = 0.1076539192... where K(k)=2E(k). - Michael Somos, Jul 21 2006

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

Example

0.1076539192264845766153234450909471905879...

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 *)

Prog

(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

Crossrefs

Cf. A072559, A073007.

Sequence in context: A372951 A334380 A101464 * A022963 A023449 A307338

Adjacent sequences: A072555 A072556 A072557 * A072559 A072560 A072561

Keyword

cons,nonn

Author

Robert G. Wilson v, Aug 03 2002

Status

approved