|
| |
|
|
A054871
|
|
a(n) = H_n(3,2) where H_n is the n-th 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_{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
|
| |
|
|
