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
Examples
0.62946502045518677183129422910723212269353...
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.
- Index entries for transcendental numbers.
Programs
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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
Comments