A006266 A continued cotangent. (Formerly M2073)

2, 14, 2786, 21624372014, 10111847525912679844192131854786, 1033930953043290626825587838528711318150300040875029341893199068078185510802565166824630504014

Offset

0,1

Comments

The next (6th) term is 280 digits long. - M. F. Hasler, Oct 06 2014

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..7

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

Formula

From Artur Jasinski, Sep 24 2008: (Start)

a(n+1) = a(n)^3 + 3*a(n) with a(0) = 2.

a(n) = round((1+sqrt(2))^(3^n)). [Corrected by M. F. Hasler, Oct 06 2014] (End)

From Peter Bala, Nov 15 2022: (Start)

a(n) = A002203(3^n).

a(n) = L(3^n,2), where L(n,x) denotes the n-th Lucas polynomial of A114525.

a(n) == 2 (mod 3).

a(n+1) == a(n) (mod 3^(n+1)) for n >= 1 (a particular case of the Gauss congruences for the companion Pell numbers).

The smallest positive residue of a(n) mod(3^n) = A271222(n).

In the ring of 3-adic integers the limit_{n -> oo} a(n) exists and is equal to A271224. Cf. A006267. (End)

Maple

a := proc(n) option remember; if n = 1 then 14 else a(n-1)^3 + 3*a(n-1) end if; end: seq(a(n), n = 1..5); # Peter Bala Nov 15 2022

Mathematica

Table[Round[(1+Sqrt[2])^(3^n)], {n, 0, 10}] (* Artur Jasinski, Sep 24 2008 *)
LucasL[3^Range[0, 7], 2] (* G. C. Greubel, Mar 25 2022 *)

Prog

(PARI) a(n, s=2)=for(i=2, n, s*=(s^2+3)); s \\ M. F. Hasler, Oct 06 2014
(Magma) [Evaluate(DicksonFirst(3^n, -1), 2): n in [0..7]]; // G. C. Greubel, Mar 25 2022
(Sage) [lucas_number2(3^n, 2, -1) for n in (0..7)] # G. C. Greubel, Mar 25 2022

Crossrefs

Cf. A002203, A006267, A114525, A145451, A145452, A271222, A271224.

Sequence in context: A277288 A296412 A296410 * A284378 A106484 A228342

Adjacent sequences: A006263 A006264 A006265 * A006267 A006268 A006269

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Edited by M. F. Hasler, Oct 06 2014

Offset corrected by G. C. Greubel, Mar 25 2022

Status

approved