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A054871 a(n) = H_n(3,2) where H_n is the n-th hyperoperator.

%I A054871 #115 Feb 16 2025 08:32:42
%S A054871 3,5,6,9,27,7625597484987
%N A054871 a(n) = H_n(3,2) where H_n is the n-th hyperoperator.
%C A054871 H_n(x,y) is defined recursively by:
%C A054871   H_0(x,y) = y+1;
%C A054871   H_1(x,0) = x;
%C A054871   H_2(x,0) = 0;
%C A054871   H_n(x,0) = 1, for n>2;
%C A054871   H_n(x,y) = H_{n-1}(x,H_n(x,y-1)), for integers n>0 and y>0.
%C A054871 Consequently:
%C A054871   H_0(x,y) = y+1  is the successor function on y;
%C A054871   H_1(x,y) = x+y  is addition;
%C A054871   H_2(x,y) = x*y  is multiplication;
%C A054871   H_3(x,y) = x^y  is exponentiation;
%C A054871   H_4(x,y) = x^^y is tetration (a height-y exponential tower x^x^x^... );
%C A054871   ...
%C A054871 Extending to negative-order hyperoperators via the recursive formula:
%C A054871   H_0(x,y) = H_{-1}(x,H_0(x,y-1)) = H_{-1}(x,y).
%C A054871 Therefore:
%C A054871   H_{-n}(x,y) = H_0(x,y), for every nonnegative n.
%C A054871 This function is an Ackermann function variant because it satisfies the recurrence relation above (see A046859).
%C A054871 Other hyperoperation notations equivalent to H_n(x,y) include:
%C A054871 Square Bracket or Box: a [n] b;
%C A054871 Conway Chain Arrows: a -> b -> n-2;
%C A054871 Knuth Up-arrow: a "up-arrow"(n-2) b;
%C A054871 Standard Caret: a ^(n-2) b.
%C A054871 Originally published as 3 agg-op-n 3 for n > 0. - _Natan Arie Consigli_, Apr 22 2015
%C A054871 Sequence can also be defined as a(0) = 3, a(1) = 5, a(n) = H_{n-1}(3,3) for n > 1. - _Natan Arie Consigli_, Apr 22 2015; edited by _Danny Rorabaugh_, Oct 18 2015
%C A054871 Before introducing the H_n notation, this sequence was named "3 agg-op-n 2, where the binary aggregation operators agg-op-n are zeration, addition, multiplication, exponentiation, superexponentiation, ..." - _Danny Rorabaugh_, Oct 14 2015
%C A054871 The next term is 3^3^...^3 (with 7625594784987 3's). - _Jianing Song_, Dec 25 2018
%D A054871 John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 60.
%H A054871 Rick Norwood, <a href="http://www.jstor.org/stable/2691346">Math. Bite: Why 2 + 2 = 2 * 2</a>, Mathematics Magazine, Vol. 71 (1998), p. 60.
%H A054871 Stephen R. Wassell, <a href="http://www.jstor.org/stable/2691082">Superexponentiation and Fixed Points of Exponential and Logarithmic Functions</a>, Mathematics Magazine, Vol. 73 (2000), pp. 111-119.
%H A054871 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/AckermannFunction.html">Ackermann Function</a> and <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H A054871 Wikipedia, <a href="http://en.wikipedia.org/wiki/Hyperoperation">Hyperoperation</a>
%H A054871 <a href="/index/Ho">Index Section Ho-Hy</a>
%e A054871 a(0) = H_0(3,2) = 2+1 = 3;
%e A054871 a(1) = H_1(3,2) = 3+2 = 5;
%e A054871 a(2) = H_2(3,2) = 3*2   = 3+3  = 6;
%e A054871 a(3) = H_3(3,2) = 3^2   = 3*3  = 9;
%e A054871 a(4) = H_4(3,2) = 3^^2  = 3^3  = 27;
%e A054871 a(5) = H_5(3,2) = 3^^^2 = 3^^3 = 3^(3^3) = 7625597484987.
%Y A054871 H_n(x,y) for various x,y: A001695 (2,n), this sequence (3,2; almost 3,3), A067652 (2,3; almost 2,4), A141044 (1,1), A175796 (n,2), A179184 (0,0), A189896 (n,n), A213619 (n,H_n(n,n)), A253855 (4,2; almost 4,4), A255176 (2,2),  A255340 (4,3), A256131 (10,2; almost 10,10), A261143 (1,2), A261146 (n,3). - _Natan Arie Consigli_ and _Danny Rorabaugh_, Oct 14-26 2015
%Y A054871 H_4(x,n) for various x: A000035 (x=0), A014221 (x=2), A014222 (x=3, shifted), A057427 (x=1).
%Y A054871 H_5(x,n) for various x: A266198 (x=2), A266199 (x=3).
%Y A054871 Cf. A254225, A254310, A257229.
%K A054871 nonn
%O A054871 0,1
%A A054871 _Walter Nissen_, May 28 2000
%E A054871 First two terms prepended by _Natan Arie Consigli_, Apr 22 2015
%E A054871 First term corrected and hyperoperator notation implemented by _Danny Rorabaugh_, Oct 14 2015
%E A054871 Definition extended to include negative n by _Natan Arie Consigli_, Oct 19 2015
%E A054871 More hyperoperator notation added by _Natan Arie Consigli_, Jan 19 2016