

A054871


a(n) = H_n(3,2) where H_n is the nth hyperoperator.


34




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_{n1}(x,H_n(x,y1)), 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 heighty exponential tower x^x^x^... );
...
Extending to negativeorder hyperoperators via the recursive formula:
H_0(x,y) = H_{1}(x,H_0(x,y1)) = 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 > n2;
Knuth Uparrow: a "uparrow"(n2) b;
Standard Caret: a ^(n2) b.
Originally published as 3 aggopn 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_{n1}(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 aggopn 2, where the binary aggregation operators aggopn 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. 111119.
Eric Weisstein's MathWorld, Ackermann Function and Power Tower
Wikipedia, Hyperoperation
Index Section HoHy


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 1426 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



