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

0,1

Comments

Cahen proved that his constant is irrational. Davison and Shallit proved that it is transcendental and computed its simple continued fraction expansion A006280. - Jonathan Sondow, Aug 17 2014

Named after the French mathematician Eugène Cahen (1865 - 1941). - Amiram Eldar, Oct 29 2020

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.7, p. 436.

Links

Table of n, a(n) for n=0..104.

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.

Formula

Equals Sum_{k >= 0} (-1)^k/(A000058(k)-1).

Equals Sum_{n>=0} 1/A000058(2*n) = 1 - Sum_{n>=0} 1/A000058(2*n+1). - Amiram Eldar, Oct 29 2020

Equals 1 + (1/2) * Sum_{n>=0} (-1)^(n+1)/A129871(n). - Bernard Schott, Apr 06 2021

Example

0.6434105462883380261...

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 *)

Prog

(PARI) C=1; 1+suminf(k=1, C+=C^2; (-1)^k/C) \\ Charles R Greathouse IV, Jul 14 2020

Crossrefs

Cf. A000058, A006279, A006280, A006281, A123180, A129871, A242724.

Sequence in context: A155044 A245634 A182618 * A199429 A235509 A346696

Adjacent sequences: A118224 A118225 A118226 * A118228 A118229 A118230

Keyword

nonn,cons

Author

Eric W. Weisstein, Apr 16 2006

Status

approved