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

Author

Benoit Cloitre, Nov 06 2002

Keywords

• cons
• easy
• nonn

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

This constant is transcendental. - Quanyu Tang, Mar 20 2025

Examples

			1.26408473530530111307959958416466949111456...
		

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).
• A. Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, Ramanujan J, Vol. 57, 2022, pp. 569-581.
• Steven Finch, Exercises in Iterational Asymptotics, arXiv:2411.16062 [math.NT], 2024. See p. 10.
• Zheng Li and Quanyu Tang, On a conjecture of Erdős and Graham about the Sylvester's sequence, arXiv:2503.12277 [math.NT], 2025. See p. 3.
• 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.

Crossrefs

Cf. A000058.

Programs

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 *)

Formula

Equals lim_{n->oo} 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