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

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

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), 419-429.

Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. 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: A340615 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