cp's OEIS Frontend

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

%I A242724 #15 Feb 16 2025 08:33:22
%S A242724 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,
%T A242724 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,
%U A242724 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
%N A242724 Decimal expansion of a constant associated with self-generating continued fractions and Cahen's constant.
%C A242724 This constant is known to be transcendental.
%C A242724 Called the "Davison-Shallit constant" by Finch (2003) and Sondow (2021). - _Amiram Eldar_, Mar 19 2024
%D A242724 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.7, p. 435.
%H A242724 Eugène Cahen, <a href="http://archive.numdam.org/ARCHIVE/NAM/NAM_1891_3_10_/NAM_1891_3_10__508_0/NAM_1891_3_10__508_0.pdf">Note sur un développement des quantités numériques, qui présente quelque analogie avec celui en fractions continues</a>, Nouvelles Annales de Mathématiques, Vol. 10 (1891), pp. 508-514. In French.
%H A242724 J. L. Davison and Jeffrey O. Shallit, <a href="https://doi.org/10.1007/BF01332350">Continued Fractions for Some Alternating Series</a>, Monatshefte für Mathematik, Vol. 111 (1991), pp. 119-126; <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN362162050_0111&amp;DMDID=DMDLOG_0013&amp;LOGID=LOG_0013&amp;PHYSID=PHYS_0126">alternative link</a>.
%H A242724 Jonathan Sondow, <a href="https://doi.org/10.1007/978-3-030-84304-5_22">Irrationality and Transcendence of Alternating Series via Continued Fractions</a>, 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; <a href="https://arxiv.org/abs/2009.14644">arXiv preprint</a>, arXiv:2009.14644 [math.NT], 2020.
%H A242724 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/CahensConstant.html">Cahen's constant</a>.
%H A242724 Wikipedia, <a href="http://en.wikipedia.org/wiki/Cahen&#39;s_constant">Cahen's constant</a>.
%H A242724 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A242724 Equals Sum_{k>=0} (-1)^k/(A006277(k)*A006277(k+1)). - _Amiram Eldar_, Mar 19 2024
%e A242724 0.62946502045518677183129422910723212269353...
%t A242724 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
%Y A242724 Cf. A006277, A006279, A006280, A006281, A118227, A123180.
%K A242724 nonn,cons
%O A242724 0,1
%A A242724 _Jean-François Alcover_, May 21 2014