%I A072449
%S A072449 1,7,5,7,9,3,2,7,5,6,6,1,8,0,0,4,5,3,2,7,0,8,8,1,9,6,3,8,2,1,8,1,3,8,5,
%T A072449 2,7,6,5,3,1,9,9,9,2,2,1,4,6,8,3,7,7,0,4,3,1,0,1,3,5,5,0,0,3,8,5,1,1,0,
%U A072449 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 1, 1,3,7,1,1,1,2,3,1,4,1,1,2,1,2,20,1,2,2,...
%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 Aaron Herschfeld, "On Infinite Radicals", American Mathematical Monthly 
               42:7 (1935), pp. 419-429.
%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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               NestedRadical.html">Nested Radical</a>
%H A072449 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               NestedRadicalConstant.html">Nested Radical Constant</a>
%e A072449 Sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ... =~ 1.757932756618004532708819638218138527653...
%t A072449 RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Range[100]]], 10, 111][[1]] 
               A New Kind Of Science
%o A072449 (PARI) (gives at least 180 correct digits) s=200; for(n=1,199,t=200-n+sqrt(s); 
               s=t);sqrt(s)
%Y A072449 Cf. A072450.
%Y A072449 Sequence in context: A163505 A021136 A143297 this_sequence A088839 A111769 
               A111513
%Y A072449 Adjacent sequences: A072446 A072447 A072448 this_sequence A072450 A072451 
               A072452
%K A072449 nonn,cons
%O A072449 1,2
%A A072449 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 01 2002