A006268 A continued cotangent. (Formerly M3141)

3, 36, 46764, 102266868132036, 1069559300034650646049671039050649693658764

Offset

0,1

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=0..4.

Jeffrey Shallit, Predictable regular continued cotangent expansions, J. Res. Nat. Bur. Standards Sect. B 80B (1976), no. 2, 285-290.

Formula

From Artur Jasinski, Oct 03 2008: (Start)

a(n+1) = a(n)^3 + 3*a(n) and a(0)=3.

a(n) = round((3/2 + sqrt(13)/2)^(3^(n - 1))). (End)

From Peter Bala, Jan 19 2022: (Start)

a(n) = (3/2 + sqrt(13)/2)^(3^(n-1)) + (3/2 - sqrt(13)/2)^(3^(n-1))

a(n) divides a(n+1) and b(n) = a(n+1)/a(n) satisfies the recurrence b(n+1) = b(n)^3 - 3*b(n-1)^2 + 3. For remarks about this recurrence see A002813.

1 + a(n)^2 = A006273(n+1). (End)

Mathematica

a = {}; k = 3; Do[AppendTo[a, k]; k = k^3 + 3 k, {n, 1, 6}]; a (* Artur Jasinski, Oct 03 2008 *)
Table[Round[N[(3/2 + Sqrt[13]/2)^(3^(n - 1)), 1000]], {n, 1, 8}] (* Artur Jasinski, Oct 03 2008 *)

Prog

(PARI) a(n) = if (n==0, 3, a(n-1)^3 + 3*a(n-1)); \\ Michel Marcus, Aug 28 2020

Crossrefs

Continued cotangents: A006267, A006266, A006269, A145180, A145181, A145182, A145183, A145184, A145185, A145186, A145187, A145188, A145189 (k = 1 to 15 with k=4 being A006267(n+1)).

Cf. A002813, A006273.

Sequence in context: A088322 A342300 A080807 * A264842 A210508 A073236

Adjacent sequences: A006265 A006266 A006267 * A006269 A006270 A006271

Keyword

nonn,easy

Author

N. J. A. Sloane.

Status

approved