A143347 Decimal expansion of the paper-folding constant, or the dragon constant.

8, 5, 0, 7, 3, 6, 1, 8, 8, 2, 0, 1, 8, 6, 7, 2, 6, 0, 3, 6, 7, 7, 9, 7, 7, 6, 0, 5, 3, 2, 0, 6, 6, 6, 0, 4, 4, 1, 1, 3, 9, 9, 4, 9, 3, 0, 8, 2, 7, 1, 0, 6, 4, 7, 7, 2, 8, 1, 6, 8, 2, 6, 1, 0, 3, 1, 8, 3, 0, 1, 5, 8, 4, 5, 9, 3, 1, 9, 4, 4, 5, 3, 4, 8, 5, 4, 5, 9, 8, 2, 6, 4, 2, 1, 9, 3, 9, 2, 3, 9, 9, 6, 0, 9, 1

Offset: 0

Author

Eric W. Weisstein, Aug 09 2008

Keywords

• nonn
• cons

Comments

Named "the Gaussian Liouville number" by Borwein and Coons (2008). - Amiram Eldar, Apr 29 2021

Examples

			0.85073618820186726036...
		

References

• Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves -- I and II, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. Reprinted in Donald E. Knuth, Selected Papers on Fun and Games, CSLI Publications, 2010, pages 571-614.
• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, section 6.8.5 Paper Folding, pages 439-440.

Links

• Joerg Arndt, Matters Computational (The Fxtbook), p. 744.
• Peter Borwein and Michael Coons, Transcendence of the Gaussian Liouville number and relatives, arXiv:0806.1694 [math.NT], 2008.
• Michael J. Coons, Some aspects of analytic number theory: parity, transcendence, and multiplicative functions, Ph.D. Thesis, Department of Mathematics, Simon Fraser University, 2009.
• Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. [Cached copy, with permission]
• J. H. Loxton, A method of Mahler in transcendence theory and some of its applications, Bulletin of the Australian Mathematical Society, Vol. 29, No. 1 (1984), pp. 127-136.
• Michel Mendès France and Alf van der Poorten, Arithmetic and Analytic Properties of Paper Folding Sequences, Bulletin of the Australian Mathematical Society, volume 24, issue 1, 1981, pages 123-131.
• A. J. van der Poorten and J. H. Loxton, Arithmetic properties of the solutions of a class of functional equations, Journal für die reine und angewandte Mathematik, Vol. 330 (1982), pp. 159-172; alternative link.
• Eric Weisstein's World of Mathematics, Paper Folding Constant.
• Index entries for sequences obtained by enumerating foldings
• Index entries for transcendental numbers

Crossrefs

Cf. A014577 (binary expansion), A034947.

Programs

Mathematica

RealDigits[ N[ Sum[ 8^2^k/(2^2^(k + 2) - 1), {k, 0, Infinity}], 110]][[1]][[1 ;; 105]] (* Jean-François Alcover, Oct 26 2012 *)

PARI

default(realprecision,510);
c=sum(k=0, 10, 1.0/( 2^(2^k) * ( 1 - 1/(2^(2^(k+2))) ) ) )
/* Joerg Arndt, Aug 28 2011 */

Formula

Equals Sum_{k>=1} A014577(k)/2^k = Sum_{k>=1} (1+A034947(k))/2^(k+1). - Amiram Eldar, Apr 29 2021