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).
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Edited by M. F. Hasler, Oct 06 2014
Offset corrected by G. C. Greubel, Mar 25 2022
STATUS
approved