

A257574


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


22



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, 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, 6, 2, 4, 0, 7, 4, 5, 4, 9, 7, 7, 1, 1, 8, 5, 9, 9, 8, 0
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OFFSET

1,1


COMMENTS

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)+...)))).
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


LINKS

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..400
A. Herschfeld, On Infinite Radicals, Amer. Math. Monthly, 42 (1935), 419429.
Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC353 to PC354. Annotated and scanned copy.


EXAMPLE

sqrt(2 + sqrt(4 + sqrt(6 + sqrt(8 + ...)))) = 2.1584768723110397656558534...


PROG

(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


CROSSREFS

Cf. A072449, A257575..A257581, A105817, A099879, A001622, A105546.
Sequence in context: A108599 A108590 A109233 * A316293 A193180 A201743
Adjacent sequences: A257571 A257572 A257573 * A257575 A257576 A257577


KEYWORD

nonn,cons


AUTHOR

N. J. A. Sloane, May 02 2015


EXTENSIONS

a(27)a(87) from Hiroaki Yamanouchi, May 03 2015
Edited by M. F. Hasler, May 01 2018


STATUS

approved



