A112845 Recurrence a(n) = a(n-1)^3 - 3*a(n-1) with a(0) = 6.

6, 198, 7761798, 467613464999866416198, 102249460387306384473056172738577521087843948916391508591105798

Offset: 0

Author

Eric W. Weisstein, Sep 21 2005

Keywords

• nonn
• easy

Comments

Identical to A006243 apart from the initial term. For some general remarks on this recurrence see A001999. - Peter Bala, Nov 13 2012

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, 629-630.
• Eric Weisstein's World of Mathematics, Pierce Expansion

Crossrefs

Cf. A006275, A006276.

Cf. A006243. - R. J. Mathar, Aug 15 2008

Cf. A001999, A219160, A219161.

Programs

Mathematica

RecurrenceTable[{a[n] == a[n - 1]^3 - 3*a[n - 1], a[0] == 6}, a, {n,
  0, 5}] (* G. C. Greubel, Dec 30 2016 *)
NestList[#^3-3#&,6,5] (* Harvey P. Dale, Jul 23 2025 *)

Formula

a(n) = -2*cos(3^n*arccos(-3)).

From Peter Bala, Nov 13 2012: (Start)

a(n) = (3 + 2*sqrt(2))^(3^n) + (3 - 2*sqrt(2))^(3^n).

Product {n = 0..inf} (1 + 2/(a(n) - 1)) = sqrt(2).

(End)