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 A105415 Decimal expansion of the Paris constant. 0
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS From Amiram Eldar, Aug 21 2020: (Start) Named after the British mathematician Richard B. Paris. 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). Also, c = Product_{k>=2} 2*phi/(phi + u(k)) (Plouffe). 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) REFERENCES Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 8. LINKS Table of n, a(n) for n=1..102. R. B. Paris, An Asymptotic Approximation Connected with the Golden Number, The American Mathematical Monthly, Vol. 94, No. 3 (1987), pp. 272-278. T. Piezas III, Golden ratio and nested radicals. Simon Plouffe, Generalized expansions of real numbers, 2006. Simon Plouffe, The Paris constant. Eric Weisstein's World of Mathematics, Paris Constant. EXAMPLE 1.098641964394156485734668917343596210873348396108... MATHEMATICA 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 *) \$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 *) CROSSREFS Cf. A001622. Sequence in context: A087044 A246168 A248585 * A346728 A197015 A239387 Adjacent sequences: A105412 A105413 A105414 * A105416 A105417 A105418 KEYWORD nonn,cons AUTHOR Eric W. Weisstein, Apr 05 2005 STATUS approved

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