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A006268 A continued cotangent.
(Formerly M3141)
13
3, 36, 46764, 102266868132036, 1069559300034650646049671039050649693658764 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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)).
Sequence in context: A088322 A342300 A080807 * A264842 A210508 A073236
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 29 10:44 EDT 2024. Contains 371268 sequences. (Running on oeis4.)