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A257574 Continued square root map applied to the sequence of positive even numbers, (2, 4, 6, 8, ...).

%I A257574 #31 May 04 2018 18:04:10
%S A257574 2,1,5,8,4,7,6,8,7,2,3,1,1,0,3,9,7,6,5,6,5,5,8,5,3,4,7,9,8,0,7,0,2,5,
%T A257574 2,4,1,6,6,9,6,9,4,4,4,0,3,5,4,2,8,6,6,7,0,3,7,5,5,0,9,6,3,4,2,1,9,4,
%U A257574 6,2,4,0,7,4,5,4,9,7,7,1,1,8,5,9,9,8,0
%N A257574 Continued square root map applied to the sequence of positive even numbers, (2, 4, 6, 8, ...).
%C A257574 The continued square root or CSR map applied to a sequence b = (b(1), b(2), b(3), ...) is the number CSR(b) := sqrt(b(1)+sqrt(b(2)+sqrt(b(3)+sqrt(b(4)+...)))).
%C A257574 Taking out a factor sqrt(2), one gets CSR(2, 4, 6, 8, ...) = sqrt(2) CSR(1, 1, 3/8, 1/32, ...) < A002193*A001622 = (sqrt(5)+1)/sqrt(2). - _M. F. Hasler_, May 01 2018
%H A257574 Hiroaki Yamanouchi, <a href="/A257574/b257574.txt">Table of n, a(n) for n = 1..400</a>
%H A257574 A. Herschfeld, <a href="http://www.jstor.org/stable/2301294">On Infinite Radicals</a>, Amer. Math. Monthly, 42 (1935), 419-429.
%H A257574 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.
%e A257574 sqrt(2 + sqrt(4 + sqrt(6 + sqrt(8 + ...)))) = 2.1584768723110397656558534...
%o A257574 (PARI) (CSR(v,s)=forstep(i=#v,1,-1,s=sqrt(v[i]+s));s); t=0;for(N=5,oo,(t==t=Str(CSR([1..2*N]*2)))&&break;print(2*N": "t)) \\ Allows to see the convergence, which is reached when length of vector ~ precision [given as number of digits]. Using Str() to avoid infinite loop when internal representation is "fluctuating". - _M. F. Hasler_, May 04 2018
%Y A257574 Cf. A072449, A257575..A257581, A105817, A099879, A001622, A105546.
%K A257574 nonn,cons
%O A257574 1,1
%A A257574 _N. J. A. Sloane_, May 02 2015
%E A257574 a(27)-a(87) from _Hiroaki Yamanouchi_, May 03 2015
%E A257574 Edited by _M. F. Hasler_, May 01 2018