A105415 Decimal expansion of the Paris constant.

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

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