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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
R. B. Paris, An Asymptotic Approximation Connected with the Golden Number, The American Mathematical Monthly, Vol. 94, No. 3 (1987), pp. 272-278.
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
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Apr 05 2005
STATUS
approved

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Last modified June 24 06:40 EDT 2024. Contains 373663 sequences. (Running on oeis4.)