cp's OEIS Frontend

A105818 Continued fraction expansion of the Fibonacci nested radical (A105817).

%I A105818 #23 Feb 16 2025 08:32:57
%S A105818 1,1,1,1,23,18,1,1,1,1,1,1,2,1,22,2,1,53,1,1,10,1,1,17,2,4,1,27,1,2,
%T A105818 422,3,3,13,12,5,28,1,3,1,2,1,3,2,4,6,6,3,5,50,1,1,6,3,2,1,118,2,1,1,
%U A105818 2,6,1,4,1,1,5,2,3,3,16,1,4,6,2,2,22,4,3,10,1,1,49,5,1,1,12,1,1,3,13,3,10,1,2
%N A105818 Continued fraction expansion of the Fibonacci nested radical (A105817).
%C A105818 The decimal expansion of this is A105817. "It was discovered by T. Vijayaraghavan that the infinite radical, sqrt( a_1 + sqrt( a_2 + sqrt ( a_3 + sqrt( a_4 + ... where a_n => 0, will converge to a limit if and only if the limit of (ln a_n)/2^n exists." [Clawson, 229; Sloane]. We know the asymptotic limit of Fibonacci numbers is Phi^n (Binet expansion) and that Phi^n < 2^n and hence that the Fibonacci Nested Radical converges.
%C A105818 Clawson misstates Vijayaraghavan's theorem. Vijayaraghavan proved that for a_n > 0, the infinite radical sqrt(a_1 + sqrt(a_2 + sqrt(a_3 + ...))) converges if and only if limsup (log a_n)/2^n < infinity. (For example, suppose a_n = 1 if n is odd, and a_n = e^2^n if n is even. Then (log a_n)/2^n = 0, 1, 0, 1, 0, 1, ... for n >= 1, so the limit does not exist. However, limsup (log a_n)/2^n = 1 and the infinite radical converges.) - _Jonathan Sondow_, Mar 25 2014
%D A105818 Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 & 229.
%D A105818 S. R. Finch, "Analysis of a Radical Expansion." Section 1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 8, 2003.
%H A105818 Jonathan M. Borwein and G. de Barra, <a href="http://www.jstor.org/stable/2324426">Nested Radicals</a>, Amer. Math. Monthly 98, 735-739, 1991.
%H A105818 J. Sondow and P. Hadjicostas, <a href="http://dx.doi.org/10.1016/j.jmaa.2006.09.081">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, J. Math. Anal. Appl., 332 (2007), 292-314; see pp. 305-306.
%H A105818 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NestedRadicalConstant.html">Nested Radical Constant.</a>
%H A105818 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tirukkannapuram_Vijayaraghavan">Tirukkannapuram Vijayaraghavan</a>
%F A105818 Sqrt(1 + Sqrt(1 + Sqrt(2 + Sqrt(3 + Sqrt(5 + ... + Sqrt(Fibonacci(n) = A000045)))).
%e A105818 1.66198246232781155796760608181513129505616756246503500829906806743...
%t A105818 f[n_] := Block[{k = n, s = 0}, While[k > 0, s = Sqrt[s + Fibonacci[k]]; k-- ]; s]; ContinuedFraction[ f[46], 95] (* _Robert G. Wilson v_, Apr 21 2005 *)
%Y A105818 Cf. A000045; A072449, A083869, A099874, A099876, A099877, A099878, A099879, A105546, A105548, A105815, A105816, A105817, A239349 for other nested radicals.
%K A105818 cofr,nonn
%O A105818 0,5
%A A105818 _Jonathan Vos Post_, Apr 21 2005
%E A105818 Offset changed by _Andrew Howroyd_, Aug 03 2024