A118227 Decimal expansion of Cahen's constant.
6, 4, 3, 4, 1, 0, 5, 4, 6, 2, 8, 8, 3, 3, 8, 0, 2, 6, 1, 8, 2, 2, 5, 4, 3, 0, 7, 7, 5, 7, 5, 6, 4, 7, 6, 3, 2, 8, 6, 5, 8, 7, 8, 6, 0, 2, 6, 8, 2, 3, 9, 5, 0, 5, 9, 8, 7, 0, 3, 0, 9, 2, 0, 3, 0, 7, 4, 9, 2, 7, 7, 6, 4, 6, 1, 8, 3, 2, 6, 1, 0, 8, 4, 8, 4, 4, 0, 8, 9, 5, 5, 5, 0, 4, 6, 3, 4, 3, 1, 9, 5, 4, 0, 5, 3
Offset: 0
Examples
0.6434105462883380261...
References
- Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.7, p. 436.
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.
- J. L. Davison and Jeffrey Shallit, Continued Fractions for Some Alternating Series, Monatsh. Math., Vol. 111, No. 2 (1991), pp. 119-126, alternative link.
- Eric Weisstein's World of Mathematics, Cahen's Constant.
- Wikipedia, Cahen's constant.
- Index entries for transcendental numbers.
Programs
-
Mathematica
a[0] = 2; a[n_] := a[n] = a[n-1]^2 - a[n-1]+1; kmax = 1; FixedPoint[ RealDigits[ Sum[(-1)^k/(a[k]-1), {k, 0, kmax += 10}], 10, 105][[1]]&, kmax] (* Jean-François Alcover, Jul 28 2011, updated Jun 19 2014 *) Most@First@RealDigits@N[x=1; 1+Sum[x=x(1+x); (-1)^k/x, {k, 1, 9}], 106] (* Oliver Seipel, Aug 25 2024, after Charles R Greathouse IV *) Most@First@RealDigits@N[x=1; 1/2+Sum[x=x(1+x)(1+x+x^2); 1/(x+1), {k, 1, 4}], 106] (* Oliver Seipel, Aug 25 2024 *) -
PARI
C=1;1+suminf(k=1,C+=C^2; (-1)^k/C) \\ Charles R Greathouse IV, Jul 14 2020
Formula
Equals Sum_{k >= 0} (-1)^k/(A000058(k)-1).
Equals 1 + (1/2) * Sum_{n>=0} (-1)^(n+1)/A129871(n). - Bernard Schott, Apr 06 2021
Comments