OFFSET
1,1
COMMENTS
Equals the reciprocal of the one-ninth constant A072558.
Named after the American mathematician Richard Steven Varga (1928-2022). - Amiram Eldar, Jun 22 2021
REFERENCES
R. S. Varga, Scientific Computation on Mathematical Problems and Conjectures, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 60, Philadelphia, PA: SIAM, 1990. See Chapter 2, pp. 23-38.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
A. J. Carpenter, A. Ruttan and R. S. Varga, Extended numerical computations on the "1/9" conjecture in rational approximation theory, in: P. R. Graves-Morris, E. B. Saff and R. S. Varga (eds.), Rational approximation and interpolation, Springer, Berlin, Heidelberg, 1984, pp. 383-411; alternative link.
Alphonse P. Magnus and Jean Meinguet, The elliptic functions and integrals of the '1/9' problem, Numerical Algorithms, Vol. 24, No. 1 (2000), pp. 117-139; alternative link.
Simon Plouffe, One-ninth constant.
Eric Weisstein's World of Mathematics, One-Ninth Constant.
EXAMPLE
9.28902549192081891875544943595174506...
MATHEMATICA
nmax=250; c = k /. FindRoot[EllipticK[k^2] == 2*EllipticE[k^2], {k, 9/10}, WorkingPrecision -> nmax]; Take[RealDigits[1/N[Exp[-Pi*(EllipticK[1 - c^2]/EllipticK[c^2])], nmax]][[1]], 200] (* G. C. Greubel, Mar 10 2018 *)
RealDigits[v /. FindRoot[4 EllipticE[InverseEllipticNomeQ[1/v]] == Pi EllipticTheta[3, 0, 1/v]^2, {v, 9, 9, 10}, WorkingPrecision -> 101]][[1]] (* Jan Mangaldan, Jun 25 2020 *)
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Robert G. Wilson v, Aug 03 2002
STATUS
approved