This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A118227 #68 Feb 16 2025 08:33:01
%S A118227 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,
%T A118227 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,
%U A118227 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
%N A118227 Decimal expansion of Cahen's constant.
%C A118227 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
%C A118227 Named after the French mathematician Eugène Cahen (1865 - 1941). - _Amiram Eldar_, Oct 29 2020
%D A118227 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.7, p. 436.
%H A118227 Eugène Cahen, <a href="http://www.numdam.org/item?id=NAM_1891_3_10__508_0">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.
%H A118227 J. L. Davison and Jeffrey Shallit, <a href="https://doi.org/10.1007/BF01332350">Continued Fractions for Some Alternating Series</a>, Monatsh. Math., Vol. 111, No. 2 (1991), pp. 119-126, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002487543">alternative link</a>.
%H A118227 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CahensConstant.html">Cahen's Constant</a>.
%H A118227 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cahen%27s_constant">Cahen's constant</a>.
%H A118227 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A118227 Equals Sum_{k >= 0} (-1)^k/(A000058(k)-1).
%F A118227 Equals Sum_{n>=0} 1/A000058(2*n) = 1 - Sum_{n>=0} 1/A000058(2*n+1). - _Amiram Eldar_, Oct 29 2020
%F A118227 Equals 1 + (1/2) * Sum_{n>=0} (-1)^(n+1)/A129871(n). - _Bernard Schott_, Apr 06 2021
%e A118227 0.6434105462883380261...
%t A118227 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 *)
%t A118227 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_ *)
%t A118227 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 *)
%o A118227 (PARI) C=1;1+suminf(k=1,C+=C^2; (-1)^k/C) \\ _Charles R Greathouse IV_, Jul 14 2020
%Y A118227 Cf. A000058, A006279, A006280, A006281, A123180, A129871, A242724.
%K A118227 nonn,cons
%O A118227 0,1
%A A118227 _Eric W. Weisstein_, Apr 16 2006