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

4, 52, 140452, 2770663499604052, 21269209556953516583554114034636483645584976452

Offset: 0

Author

Peter Bala, Nov 13 2012

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A001999, A112845, A219161.

Programs

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 *)

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).