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%I #36 Jun 22 2021 12:30:09
%S 1,2,6,4,0,8,4,7,3,5,3,0,5,3,0,1,1,1,3,0,7,9,5,9,9,5,8,4,1,6,4,6,6,9,
%T 4,9,1,1,1,4,5,6,0,1,7,9,2,0,9,0,6,5,5,3,3,1,5,3,4,5,6,4,1,9,9,0,7,7,
%U 5,9,0,1,6,3,6,2,9,5,1,6,0,1,4,2,2,6,3,9,0,9,2,6,8,3,9,8,5,1,5,0,4,8,0,3,3
%N Decimal expansion of Vardi constant arising in the Sylvester sequence.
%C Vardi showed A000058(n) = floor(c^(2^(n+1))+1/2) where c=1.26408473...
%C Named after the Canadian mathematician Ilan Vardi (b. 1957). - _Amiram Eldar_, Jun 22 2021
%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 443-448.
%D Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, exercise 4.37, p. 518.
%D Ilan Vardi, Computational Recreations in Mathematica, Addison-Wesley, 1991, pp. 82-89.
%H Alfred Vaino Aho and N. J. A. Sloane, <a href="https://www.fq.math.ca/Scanned/11-4/aho-a.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437; <a href="http://neilsloane.com/doc/doubly.html">alternative link</a>. See p. 435.
%H Matthew Brendan Crawford, <a href="https://vtechworks.lib.vt.edu/handle/10919/90573">On the Number of Representations of One as the Sum of Unit Fractions</a>, Master's Thesis, Virginia Polytechnic Institute and State University (2019).
%H Benjamin Nill, <a href="https://doi.org/10.1007/s00454-006-1299-y">Volume and lattice points of reflexive simplices</a>, Discrete & Computational Geometry, Vol. 37, No. 2 (2007), pp. 301-320; <a href="https://arxiv.org/abs/math/0412480">arXiv preprint</a>, arXiv:math/0412480 [math.AG], 2004-2007.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SylvestersSequence.html">Sylvester's Sequence</a>.
%F Equals lim_{n->infinity} A000058(n)^(1/2^(n+1)). - _Robert FERREOL_, Feb 15 2019
%F Equals sqrt((3/2) * Product_{k>=0} (1 + 1/(2*A000058(k)-1)^2)^(1/2^(k+1))). - _Amiram Eldar_, Jun 22 2021
%e 1.26408473530530111307959958416466949111456...
%t digits = 105; For[c = 2; olds = -1; s = 0; j = 1, RealDigits[olds, 10, digits+5] != RealDigits[s, 10, digits+5], j++; c = c^2-c+1, olds = s; s = s + 2^(-j-1)*Log[1+(2*c-1)^-2] // N[#, digits+5]&]; chi = Sqrt[6]/2*Exp[s]; RealDigits[chi, 10, digits] // First (* _Jean-François Alcover_, Jun 05 2014 *)
%Y Cf. A000058.
%K cons,easy,nonn
%O 1,2
%A _Benoit Cloitre_, Nov 06 2002
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