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%I #32 Oct 31 2019 11:41:17
%S 1,6,6,1,9,8,2,4,6,2,3,2,7,8,1,1,5,5,7,9,6,7,6,0,6,0,8,1,8,1,5,1,3,1,
%T 2,9,5,0,5,6,1,6,7,5,6,2,4,6,5,0,3,5,0,0,8,2,9,9,0,6,8,0,6,7,4,3,0,6,
%U 2,9,7,2,3,5,9,8,9,5,7,3,8,1,0,8,1,7,1,6,7,0,4,1,1,0,8,4,9,2,6,6,6,9,2,2,5
%N Decimal expansion of the Fibonacci nested radical.
%C The continued fraction expression of this is A105818. "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 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 Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 & 229.
%D S. R. Finch, "Analysis of a Radical Expansion." Section 1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 8, 2003.
%H Chai Wah Wu, <a href="/A105817/b105817.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 Herman P. Robinson, <a href="/A257574/a257574.pdf">The CSR Function</a>, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.
%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NestedRadicalConstant.html">Nested Radical Constant.</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tirukkannapuram_Vijayaraghavan">Tirukkannapuram Vijayaraghavan</a>
%F Sqrt(1 + sqrt(1 + sqrt(2 + sqrt(3 + sqrt(5 + ... + sqrt(Fibonacci(n)=A000045)))).
%e 1.66198246232781155796760608181513129505616756246503500829906806743...
%t RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Fibonacci[ Range[50]]]], 10, 111][[1]] (* _Robert G. Wilson v_, Apr 21 2005 *)
%Y Cf. A000045; A072449, A083869, A099874, A099876, A099877, A099878, A099879, A105546, A105548, A105815, A105816, A105818, A239349 for other nested radicals.
%Y Cf. A151558.
%K cons,nonn
%O 1,2
%A _Jonathan Vos Post_, Apr 21 2005
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