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A173566 a(n+1) = a(n)^a(n), with a(1) = 2.

%I A173566 #45 Jul 08 2024 11:12:34
%S A173566 2,4,256
%N A173566 a(n+1) = a(n)^a(n), with a(1) = 2.
%C A173566 The next term, a(4), is 2^2048, with 617 digits.
%C A173566 From _Natan Arie Consigli_, Dec 01 2015: (Start)
%C A173566 Possible other sequence with the same first three entries:
%C A173566 a(1) = 2;
%C A173566 a(2) = Triangle(2);
%C A173566 a(3) = Square(2);
%C A173566 a(4) = Pentagon(2);
%C A173566 etc., where, in Steinhaus-Moser notation,
%C A173566 Triangle(n) = n^n;
%C A173566 Square(n) = Triangle(Triangle...(n)...) (with n inside n nested triangles);
%C A173566 Pentagon(n) = Square(Square...(n)...)(with n inside n nested squares);
%C A173566 etc.
%C A173566 Start with a(1) = 2, a(2) = triangle(2) = 4, a(3) = square(2) = 256, a(4) = pentagon(4) = 256^^256 (power tower of 256s with height 256).
%C A173566 (End)
%H A173566 Michel Marcus, <a href="/A173566/b173566.txt">Table of n, a(n) for n = 1..4</a>
%H A173566 Wikipedia, <a href="https://en.wikipedia.org/wiki/Steinhaus-Moser_notation">Steinhaus-Moser notation</a>.
%e A173566 a(3) = square(2) = triangle(triangle(2)) = triangle(2^2) = 4^4 = 256.
%e A173566 a(4) = 2^2048.
%e A173566 a(5) = 2^(2^2059).
%t A173566 RecurrenceTable[{a[1] == 2, a[n] == a[n - 1]^a[n - 1]}, a, {n, 4}] (* _Vincenzo Librandi_, Dec 17 2015 *)
%o A173566 (Magma) [n le 1 select 2 else Self(n-1)^Self(n-1): n in [1..4]]; // _Vincenzo Librandi_, Dec 17 2015
%Y A173566 Cf. A030798 ("preceding term"), A054874 (log base 2).
%K A173566 nonn,bref
%O A173566 1,1
%A A173566 _Franklin T. Adams-Watters_, Aug 03 2011