

A105818


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


1



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, 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, 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
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OFFSET

1,5


COMMENTS

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.
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


REFERENCES

Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 & 229.
S. R. Finch, "Analysis of a Radical Expansion." Section 1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 8, 2003.


LINKS

Table of n, a(n) for n=1..95.
Jonathan M. Borwein and G. de Barra, Nested Radicals, Amer. Math. Monthly 98, 735739, 1991.
J. Sondow and P. Hadjicostas, The generalizedEulerconstant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., 332 (2007), 292314; see pp. 305306.
Eric Weisstein's World of Mathematics, Nested Radical Constant.
Wikipedia, Tirukkannapuram Vijayaraghavan


FORMULA

Sqrt(1 + Sqrt(1 + Sqrt(2 + Sqrt(3 + Sqrt(5 + ... + Sqrt(Fibonacci(n) = A000045)))).


EXAMPLE

1.66198246232781155796760608181513129505616756246503500829906806743...


MATHEMATICA

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 *)


CROSSREFS

Cf. A000045; A072449, A083869, A099874, A099876, A099877, A099878, A099879, A105546, A105548, A105815, A105816, A105817, A239349 for other nested radicals.
Sequence in context: A217894 A204598 A160436 * A085450 A077146 A077576
Adjacent sequences: A105815 A105816 A105817 * A105819 A105820 A105821


KEYWORD

cofr,nonn


AUTHOR

Jonathan Vos Post, Apr 21 2005


STATUS

approved



