A242724 Decimal expansion of a constant associated with self-generating continued fractions and Cahen's constant.

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

Offset

0,1

Comments

This constant is known to be transcendental.

Called the "Davison-Shallit constant" by Finch (2003) and Sondow (2021). - Amiram Eldar, Mar 19 2024

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.7, p. 435.

Links

Table of n, a(n) for n=0..99.

Eugène Cahen, Note sur un développement des quantités numériques, qui présente quelque analogie avec celui en fractions continues, Nouvelles Annales de Mathématiques, Vol. 10 (1891), pp. 508-514. In French.

J. L. Davison and Jeffrey O. Shallit, Continued Fractions for Some Alternating Series, Monatshefte für Mathematik, Vol. 111 (1991), pp. 119-126; alternative link.

Jonathan Sondow, Irrationality and Transcendence of Alternating Series via Continued Fractions, in: A. Bostan and K. Raschel (eds.), Transcendence in Algebra, Combinatorics, Geometry and Number Theory, TRANS 2019. Springer Proceedings in Mathematics & Statistics, Vol. 373, Springer, Cham, 2021; arXiv preprint, arXiv:2009.14644 [math.NT], 2020.

Eric Weisstein's MathWorld, Cahen's constant.

Wikipedia, Cahen's constant.

Index entries for transcendental numbers.

Formula

Equals Sum_{k>=0} (-1)^k/(A006277(k)*A006277(k+1)). - Amiram Eldar, Mar 19 2024

Example

0.62946502045518677183129422910723212269353...

Mathematica

digits = 100; Clear[q, s]; q[n_] := q[n] = q[n - 2]*(q[n-1] + 1); q[0] = q[1] = 1; s[k_] := s[k] = Sum[(-1)^j/(q[j]*q[j+1]), {j, 0, k}] // N[#, digits+5]&; s[dk = 5]; s[k = 2*dk]; While[RealDigits[s[k], 10, digits] != RealDigits[s[k - dk], 10, digits], Print["k = ", k]; k = k + dk]; RealDigits[s[k], 10, digits] // First

Crossrefs

Cf. A006277, A006279, A006280, A006281, A118227, A123180.

Sequence in context: A351215 A215261 A132826 * A100123 A228040 A153633

Adjacent sequences: A242721 A242722 A242723 * A242725 A242726 A242727

Keyword

nonn,cons

Author

Jean-François Alcover, May 21 2014

Status

approved