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), 644-646.
N. J. Fine, Infinite products for k-th 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
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Nov 13 2012
STATUS
approved