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

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

Offset: 0

Author

Jonathan Vos Post, Apr 21 2005

Keywords

• cofr
• nonn

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

Examples

			1.66198246232781155796760608181513129505616756246503500829906806743...
		

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

• Jonathan M. Borwein and G. de Barra, Nested Radicals, Amer. Math. Monthly 98, 735-739, 1991.
• J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., 332 (2007), 292-314; see pp. 305-306.
• Eric Weisstein's World of Mathematics, Nested Radical Constant.
• Wikipedia, Tirukkannapuram Vijayaraghavan

Crossrefs

Cf. A000045; A072449, A083869, A099874, A099876, A099877, A099878, A099879, A105546, A105548, A105815, A105816, A105817, A239349 for other nested radicals.

Programs

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 *)

Formula

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

Extensions

Offset changed by Andrew Howroyd, Aug 03 2024