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%I M3141 #22 Jan 20 2022 10:29:44
%S 3,36,46764,102266868132036,
%T 1069559300034650646049671039050649693658764
%N A continued cotangent.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Jeffrey Shallit, <a href="http://archive.org/details/jresv80Bn2p285">Predictable regular continued cotangent expansions</a>, J. Res. Nat. Bur. Standards Sect. B 80B (1976), no. 2, 285-290.
%F From _Artur Jasinski_, Oct 03 2008: (Start)
%F a(n+1) = a(n)^3 + 3*a(n) and a(0)=3.
%F a(n) = round((3/2 + sqrt(13)/2)^(3^(n - 1))). (End)
%F From _Peter Bala_, Jan 19 2022: (Start)
%F a(n) = (3/2 + sqrt(13)/2)^(3^(n-1)) + (3/2 - sqrt(13)/2)^(3^(n-1))
%F 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.
%F 1 + a(n)^2 = A006273(n+1). (End)
%t a = {}; k = 3; Do[AppendTo[a, k]; k = k^3 + 3 k, {n, 1, 6}]; a (* _Artur Jasinski_, Oct 03 2008 *)
%t Table[Round[N[(3/2 + Sqrt[13]/2)^(3^(n - 1)), 1000]], {n, 1, 8}] (* _Artur Jasinski_, Oct 03 2008 *)
%o (PARI) a(n) = if (n==0, 3, a(n-1)^3 + 3*a(n-1)); \\ _Michel Marcus_, Aug 28 2020
%Y 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)).
%Y Cf. A002813, A006273.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_.
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