A073007 Decimal expansion of Varga's constant.
9, 2, 8, 9, 0, 2, 5, 4, 9, 1, 9, 2, 0, 8, 1, 8, 9, 1, 8, 7, 5, 5, 4, 4, 9, 4, 3, 5, 9, 5, 1, 7, 4, 5, 0, 6, 1, 0, 3, 1, 6, 9, 4, 8, 6, 7, 7, 5, 0, 1, 2, 4, 4, 0, 8, 2, 3, 9, 7, 0, 0, 6, 1, 4, 2, 1, 7, 2, 9, 3, 7, 5, 2, 4, 7, 2, 8, 6, 5, 0, 7, 0, 7, 0, 5, 2, 4, 1, 5, 8, 7, 0, 6, 1, 4, 2, 4, 7, 1, 4, 4
Offset: 1
Examples
9.28902549192081891875544943595174506...
References
- Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 4.5, p. 260.
- 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.
Crossrefs
Cf. A072558.
Programs
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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 *)
Comments