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A242724
Decimal expansion of a constant associated with self-generating continued fractions and Cahen's constant.
3
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
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.
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
KEYWORD
nonn,cons
AUTHOR
STATUS
approved