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A054871 a(n) = H_n(3,2) where H_n is the n-th hyperoperator. 35
3, 5, 6, 9, 27, 7625597484987 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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.
Originally published as 3 agg-op-n 3 for n > 0. - Natan Arie Consigli, Apr 22 2015
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
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
REFERENCES
John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 60.
LINKS
Rick Norwood, Math. Bite: Why 2 + 2 = 2 * 2, Mathematics Magazine, Vol. 71 (1998), p. 60.
Stephen R. Wassell, Superexponentiation and Fixed Points of Exponential and Logarithmic Functions, Mathematics Magazine, Vol. 73 (2000), pp. 111-119.
Eric Weisstein's MathWorld, Ackermann Function and Power Tower
Wikipedia, Hyperoperation
EXAMPLE
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.
CROSSREFS
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
H_4(x,n) for various x: A000035 (x=0), A014221 (x=2), A014222 (x=3, shifted), A057427 (x=1).
H_5(x,n) for various x: A266198 (x=2), A266199 (x=3).
Sequence in context: A102606 A102372 A095829 * A248644 A242197 A283051
KEYWORD
nonn
AUTHOR
Walter Nissen, May 28 2000
EXTENSIONS
First two terms prepended by Natan Arie Consigli, Apr 22 2015
First term corrected and hyperoperator notation implemented by Danny Rorabaugh, Oct 14 2015
Definition extended to include negative n by Natan Arie Consigli, Oct 19 2015
More hyperoperator notation added by Natan Arie Consigli, Jan 19 2016
STATUS
approved

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Last modified March 1 15:29 EST 2024. Contains 370440 sequences. (Running on oeis4.)