OFFSET
1,3
COMMENTS
From Amiram Eldar, Aug 21 2020, Oct 05 2024: (Start)
Named after the British mathematician Richard Bruce Paris (1946-2022).
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
Steven Finch, Iterated Radical Expansions and Convergence, arXiv:2410.02114 [math.NT], 2024. See p. 11.
R. B. Paris, An Asymptotic Approximation Connected with the Golden Number, The American Mathematical Monthly, Vol. 94, No. 3 (1987), pp. 272-278.
Tito 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
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Apr 05 2005
STATUS
approved