OFFSET
1,2
COMMENTS
Vardi showed A000058(n) = floor(c^(2^(n+1))+1/2) where c=1.26408473...
Named after the Canadian mathematician Ilan Vardi (b. 1957). - Amiram Eldar, Jun 22 2021
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 443-448.
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, exercise 4.37, p. 518.
Ilan Vardi, Computational Recreations in Mathematica, Addison-Wesley, 1991, pp. 82-89.
LINKS
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Matthew Brendan Crawford, On the Number of Representations of One as the Sum of Unit Fractions, Master's Thesis, Virginia Polytechnic Institute and State University (2019).
Steven Finch, Exercises in Iterational Asymptotics, arXiv:2411.16062 [math.NT], 2024. See p. 10.
Benjamin Nill, Volume and lattice points of reflexive simplices, Discrete & Computational Geometry, Vol. 37, No. 2 (2007), pp. 301-320; arXiv preprint, arXiv:math/0412480 [math.AG], 2004-2007.
Eric Weisstein's World of Mathematics, Sylvester's Sequence.
FORMULA
Equals lim_{n->infinity} A000058(n)^(1/2^(n+1)). - Robert FERREOL, Feb 15 2019
Equals sqrt((3/2) * Product_{k>=0} (1 + 1/(2*A000058(k)-1)^2)^(1/2^(k+1))). - Amiram Eldar, Jun 22 2021
EXAMPLE
1.26408473530530111307959958416466949111456...
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
AUTHOR
Benoit Cloitre, Nov 06 2002
STATUS
approved