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

3, 5, 6, 9, 27, 7625597484987

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

Table of n, a(n) for n=0..5.

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

Index Section Ho-Hy

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).

Cf. A254225, A254310, A257229.

Sequence in context: A102606 A102372 A095829 * A248644 A242197 A283051

Adjacent sequences: A054868 A054869 A054870 * A054872 A054873 A054874

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