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A054871
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a(n) = H_n(3,2) where H_n is the n-th hyperoperator.
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35
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OFFSET
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0,1
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COMMENTS
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H_n(x,y) is defined recursively by:
H_0(x,y) = y+1;
H_1(x,0) = x;
H_2(x,0) = 0;
H_n(x,0) = 1, for n>2;
H_n(x,y) = H_{n-1}(x,H_n(x,y-1)), for integers n>0 and y>0.
Consequently:
H_0(x,y) = y+1 is the successor function on y;
H_1(x,y) = x+y is addition;
H_2(x,y) = x*y is multiplication;
H_3(x,y) = x^y is exponentiation;
H_4(x,y) = x^^y is tetration (a height-y exponential tower x^x^x^... );
...
Extending to negative-order hyperoperators via the recursive formula:
H_0(x,y) = H_{-1}(x,H_0(x,y-1)) = H_{-1}(x,y).
Therefore:
H_{-n}(x,y) = H_0(x,y), for every nonnegative n.
This function is an Ackermann function variant because it satisfies the recurrence relation above (see A046859).
Other hyperoperation notations equivalent to H_n(x,y) include:
Square Bracket or Box: a [n] b;
Conway Chain Arrows: a -> b -> n-2;
Knuth Up-arrow: a "up-arrow"(n-2) b;
Standard Caret: a ^(n-2) b.
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
The next term is 3^3^...^3 (with 7625594784987 3's). - Jianing Song, Dec 25 2018
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REFERENCES
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John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 60.
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LINKS
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EXAMPLE
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a(0) = H_0(3,2) = 2+1 = 3;
a(1) = H_1(3,2) = 3+2 = 5;
a(2) = H_2(3,2) = 3*2 = 3+3 = 6;
a(3) = H_3(3,2) = 3^2 = 3*3 = 9;
a(4) = H_4(3,2) = 3^^2 = 3^3 = 27;
a(5) = H_5(3,2) = 3^^^2 = 3^^3 = 3^(3^3) = 7625597484987.
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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First term corrected and hyperoperator notation implemented by Danny Rorabaugh, Oct 14 2015
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STATUS
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approved
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