A076393 Decimal expansion of Vardi constant arising in the Sylvester sequence.

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, 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, 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

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

Table of n, a(n) for n=1..105.

Alfred Vaino Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437; alternative link. See p. 435.

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).

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

Cf. A000058.

Sequence in context: A077750 A332253 A247493 * A054674 A186503 A213654

Adjacent sequences: A076390 A076391 A076392 * A076394 A076395 A076396

Keyword

cons,easy,nonn

Author

Benoit Cloitre, Nov 06 2002

Status

approved