cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

3, 5, 6, 9, 27, 7625597484987
Offset: 0

Views

Author

Walter Nissen, May 28 2000

Keywords

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

Examples

			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.

References

  • John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 60.

Links

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.

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

A189896 Weak Ackermann numbers: H_n(n,n) where H_n is the n-th hyperoperator.

Original entry on oeis.org

1, 2, 4, 27
Offset: 0

Views

Author

Max Sills, Apr 30 2011

Keywords

Comments

The next term, a(4), has about 8*10^153 decimal digits. - Charles R Greathouse IV, Nov 15 2022

Examples

			a(0) = succ(0) = 0 + 1 = 1, because the zeroth hyperoperation is successor.
a(1) = 1 + 1 = 2, because the first hyperoperation is addition.
a(2) = 2 * 2 = 4, because the second hyperoperation is multiplication.
a(3) = 3^3 = 27, because the third hyperoperation is exponentiation.
a(4) = 4^4^4^4 = 4^(4^(4^4)) = 4^(4^256), because the fourth hyperoperation is tetration. The term is too big to be included: log_2(a(4)) = 2^513.

Links

Crossrefs

For H_n(x,x) with fixed x, cf. A054871 (x=3, shifted), A141044 (x=1), A253855 (x=4, shifted), A255176 (x=2), A256131 (x=10, shifted). - Danny Rorabaugh, Oct 20 2015
Cf. A271553 ( H_n-1(n,n) ). - Natan Arie Consigli, Apr 10 2016

Formula

a(n) = H_n(n, n), where H_n the hyperoperation indexed by n.

Extensions

"Weak" added to definition by Natan Arie Consigli, Apr 18 2015

A175796 H_n(n, 2) where H_c(a, b) is the hyperoperation function with operator c.

Original entry on oeis.org

3, 3, 4, 9, 256
Offset: 0

Views

Author

Grant Garcia, Sep 06 2010

Keywords

Examples

			a(0) = H_0(0, 2) = 2 + 1 = 3
a(1) = H_1(1, 2) = 1 + 2 = 3
a(2) = H_2(2, 2) = 2 * 2 = 4
a(3) = H_3(3, 2) = 3 ^ 2 = 9
a(4) = H_4(4, 2) = 4 ^^ 2 = 4 ^ 4 = 256
a(5) = H_5(5, 2) = 5 ^^^ 2 = 5 ^^ 5 = 5 ^ 5 ^ 5 ^ 5 ^ 5 =~ 10 ^ (10 ^ (10 ^ (2184.1257...)))

Links

Crossrefs

For H_n(x,2) with fixed x, cf. A054871 (x=3), A253855 (x=4), A255176 (x=2), A256131 (x=10), A261143 (x=1). - Danny Rorabaugh, Oct 20 2015

Programs

  • Python
    def H(a, b, c):
    if c == 0: return b + 1
    if c == 1 and b == 0: return a
    if c == 2 and b == 0: return 0
    if c >= 3 and b == 0: return 1
    return H(a, H(a, b - 1, c), c - 1)
for n in range(5): print(H(n, 2, n))

Formula

a(n) = H_n(n, 2)
H_c(a, b) = {b + 1 if c = 0; a if c = 1, b = 0; 0 if c = 2, b = 0; 1 if c >= 3, b = 0; H_{c-1}(a, H_c(a, b - 1)) otherwise}
Showing 1-3 of 3 results.

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