

A219160


Recurrence equation a(n+1) = a(n)^3  3*a(n) with a(0) = 4.


7




OFFSET

0,1


COMMENTS

For some general remarks on this recurrence see A001999.


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..6
E. B. Escott, Rapid method for extracting a square root, Amer. Math. Monthly, 44 (1937), 644646.
N. J. Fine, Infinite products for kth roots, Amer. Math. Monthly Vol. 84, No. 8, Oct. 1977.


FORMULA

a(n) = (2 + sqrt(3))^(3^n) + (2  sqrt(3))^(3^n).
Product {n = 0..inf} (1 + 2/(a(n)  1)) = sqrt(3). The rate of convergence is cubic. Fine remarks that taking the first twelve factors of the product would give well over 300,000 correct decimals for sqrt(3).


MATHEMATICA

RecurrenceTable[{a[n] == a[n  1]^3  3*a[n  1], a[0] == 4}, a, {n,
0, 5}] (* G. C. Greubel, Dec 30 2016 *)


CROSSREFS

Cf. A001999, A112845, A219161.
Sequence in context: A327373 A193914 A288490 * A111034 A214367 A241872
Adjacent sequences: A219157 A219158 A219159 * A219161 A219162 A219163


KEYWORD

nonn,easy


AUTHOR

Peter Bala, Nov 13 2012


STATUS

approved



