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

A329684 Number of excursions of length n with Motzkin-steps forbidding all consecutive steps of length 2 except UD and HH.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 0

Views

Author

Valerie Roitner, Nov 29 2019

Keywords

Comments

The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). An excursion is a path starting at (0,0), ending on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.
This sequence is periodic with a pre-period of length 3 (namely 1, 1, 2) and a period of length 1 (namely 1).
Decimal expansion of 1009/9000. - Elmo R. Oliveira, Jun 16 2024

Examples

			a(2)=2 since UD and HH are allowed. For n different from 2, only the excursion H^n is allowed.

Links

Crossrefs

Cf. A329680, A329682, A329683.
Essentially the same as A294619, A261143 and A141044.

Programs

  • Mathematica
    PadRight[{1, 1, 2}, 100, 1] (* Paolo Xausa, Aug 28 2024 *)

Formula

G.f.: (1+t^2-t^3)/(1-t).
For n >= 0, a(2) = 2, otherwise a(n) = 1. - Elmo R. Oliveira, Jun 16 2024

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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