A219160 Recurrence equation a(n+1) = a(n)^3 - 3*a(n) with a(0) = 4.

4, 52, 140452, 2770663499604052, 21269209556953516583554114034636483645584976452

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

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