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%I #24 Sep 26 2023 09:50:15
%S 1,0,9,8,6,4,1,9,6,4,3,9,4,1,5,6,4,8,5,7,3,4,6,6,8,9,1,7,3,4,3,5,9,6,
%T 2,1,0,8,7,3,3,4,8,3,9,6,1,0,8,2,9,7,1,6,7,2,1,1,8,3,3,0,5,3,2,7,8,7,
%U 1,9,8,9,2,0,4,3,5,3,1,3,3,2,4,8,9,9,2,8,8,9,5,5,2,4,7,9,9,9,4,6,6,4
%N Decimal expansion of the Paris constant.
%C From _Amiram Eldar_, Aug 21 2020: (Start)
%C Named after the British mathematician Richard B. Paris.
%C Let u(k) a sequence of real numbers defined by u(1) = 1 and u(k) = sqrt(1 + u(k-1)) for k > 1. Then lim_{k->oo} u(k) = phi (A001622), and phi - u(k) ~ 2*c/(2*phi)^k as k -> oo, where c is this constant (Paris, 1987).
%C Also, c = Product_{k>=2} 2*phi/(phi + u(k)) (Plouffe).
%C Also, c = phi * F(1/phi), where F is the analytic solution to the functional equation F(x) = 2 * phi * F(phi - sqrt(phi^2 - x), for |x| < phi^2, with F(0) = 0 and F'(0) = 1 (Finch, 2003). (End)
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 8.
%H R. B. Paris, <a href="https://www.jstor.org/stable/2323395">An Asymptotic Approximation Connected with the Golden Number</a>, The American Mathematical Monthly, Vol. 94, No. 3 (1987), pp. 272-278.
%H T. Piezas III, <a href="https://sites.google.com/view/tpiezas/0014-article-4-golden-ratio-and-nested-radicals">Golden ratio and nested radicals</a>.
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/gendev/986419.html">Generalized expansions of real numbers</a>, 2006.
%H Simon Plouffe, <a href="http://plouffe.fr/simon/constants/paris.txt">The Paris constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ParisConstant.html">Paris Constant</a>.
%e 1.098641964394156485734668917343596210873348396108...
%t ParisC = Catch[ For[ lastc = 0; c = 1; phi = 1; n = 2, True, n++, phi = N[Sqrt[1 + phi], 110]; c = c*2*GoldenRatio / (GoldenRatio + phi); If[ c - lastc < 10^-105, Throw[c], lastc = c]]]; RealDigits[ ParisC ][[1]][[1 ;; 102]] (* _Jean-François Alcover_, Oct 26 2012 *)
%t $MaxExtraPrecision = 1000; Take[RealDigits[SequenceLimit[N[Table[(GoldenRatio - Nest[Sqrt[1 + #] &, 0, n]) (2 GoldenRatio)^n / 2, {n, 200}], 300]]][[1]], 200] (* _Vladimir Reshetnikov_, Nov 18 2015 *)
%Y Cf. A001622.
%K nonn,cons
%O 1,3
%A _Eric W. Weisstein_, Apr 05 2005
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