cp's OEIS Frontend

A072449 Decimal expansion of the limit of the nested radical sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ... )))).

%I A072449 #45 Feb 16 2025 08:32:46
%S A072449 1,7,5,7,9,3,2,7,5,6,6,1,8,0,0,4,5,3,2,7,0,8,8,1,9,6,3,8,2,1,8,1,3,8,
%T A072449 5,2,7,6,5,3,1,9,9,9,2,2,1,4,6,8,3,7,7,0,4,3,1,0,1,3,5,5,0,0,3,8,5,1,
%U A072449 1,0,2,3,2,6,7,4,4,4,6,7,5,7,5,7,2,3,4,4,5,5,4,0,0,0,2,5,9,4,5,2,9,7,0,9,3
%N A072449 Decimal expansion of the limit of the nested radical sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ... )))).
%C A072449 Herschfeld calls this the Kasner number, after Edward Kasner. - _Charles R Greathouse IV_, Dec 30 2008
%C A072449 No closed-form expression is known for this constant.
%C A072449 "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 (log a_n)/2^n exists" - Clawson, p. 229. Obviously if a_n = n, the limit of (log a_n) / 2^n as n -> infinity is 0.
%C A072449 The continued fraction is A072450.
%C A072449 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 A072449 Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 & 229.
%D A072449 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.1.
%D A072449 David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, London, England, 1997, page 30.
%D A072449 Stephen Wolfram, "A New Kind Of Science," Wolfram Media, 2002, page 915.
%H A072449 Chai Wah Wu, <a href="/A072449/b072449.txt">Table of n, a(n) for n = 1..10000</a>
%H A072449 Geoffrey B. Campbell and A. Zujev, <a href="http://zujev.physics.ucdavis.edu/papers/Some_left_nested_radicals.pdf">Some left nested radicals</a>, Preprint 2016.
%H A072449 A. Herschfeld, <a href="http://www.jstor.org/stable/2301294">On Infinite Radicals</a>, Amer. Math. Monthly, 42 (1935), 419-429.
%H A072449 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 A072449 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 A072449 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NestedRadical.html">Nested Radical</a>.
%H A072449 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NestedRadicalConstant.html">Nested Radical Constant</a>.
%H A072449 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tirukkannapuram_Vijayaraghavan">Tirukkannapuram Vijayaraghavan</a>.
%e A072449 1.757932756618004532708819638218138527653...
%t A072449 RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Range[100]]], 10, 111][[1]] (* A New Kind Of Science *)
%o A072449 (PARI) s=200; for(n=1,199,t=200-n+sqrt(s); s=t);sqrt(s) \\ gives at least 180 correct digits
%Y A072449 Cf. A072450, A239349.
%K A072449 nonn,cons
%O A072449 1,2
%A A072449 _Robert G. Wilson v_, Aug 01 2002