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%I #53 Apr 09 2021 02:33:45
%S 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 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 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 Decimal expansion of Cahen's constant.
%C 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 Named after the French mathematician Eugène Cahen (1865 - 1941). - _Amiram Eldar_, Oct 29 2020
%H 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CahensConstant.html">Cahen's Constant</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cahen%27s_constant">Cahen's constant</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Sum_{k >= 0} (-1)^k/(A000058(k)-1).
%F Equals Sum_{n>=0} 1/A000058(2*n) = 1 - Sum_{n>=0} 1/A000058(2*n+1). - _Amiram Eldar_, Oct 29 2020
%F Equals 1 + (1/2) * Sum_{n>=0} (-1)^(n+1)/A129871(n). - _Bernard Schott_, Apr 06 2021
%e 0.6434105462883380261...
%t 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 *)
%o (PARI) C=1;1+suminf(k=1,C+=C^2; (-1)^k/C) \\ _Charles R Greathouse IV_, Jul 14 2020
%Y Cf. A000058, A006279, A006280, A006281, A123180, A129871, A242724.
%K nonn,cons
%O 0,1
%A _Eric W. Weisstein_, Apr 16 2006
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