

A054871


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


35




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.
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



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


KEYWORD

nonn


AUTHOR



EXTENSIONS

First term corrected and hyperoperator notation implemented by Danny Rorabaugh, Oct 14 2015


STATUS

approved



