OFFSET

1,1

COMMENTS

Suppose that S is a set of real numbers. An S-power-tower, t, is a number t = x(1)^x(2)^...^x(k), where k >= 1 and x(i) is in S for i = 1..k. We represent t by (x(1), x(2), ..., x(k)), which for k > 1 is defined as (x(1), (x(2), ..., x(k))); (2,3,2) means 2^9. The number k is the *height* of t. If every element of S exceeds 1 and all the power towers are ranked in increasing order, the position of each in the resulting sequence is its *rank*.

In the following guide, "tower" means "power-tower", and t(n) denotes the n-th {2,3}-tower, represented as (x(1), x(2), ..., x(k)).

A299229: sequence of all {2,3}-towers, ranked, concatenated

A299230: a(n) = height of t(n)

A299231: all n such that t(n) has x(1) = 2

A299232: all n such that t(n) has x(1) = 3

A299233: all n such that t(n) has x(k) = 2

A299234: all n such that t(n) has x(k) = 3

A299235: a(n) = number of 2's in t(n)

A299236: a(n) = number of 3's in t(n)

A299237: a(n) = m satisfying t(m) = reversal of t(n)

A299238; a(n) = m satisfying t(m) = 5 - t(n)

A299239: all n such that t(n) is a palindrome

A299240: ranks of all t[n] in which #2's > #3's

A299241: ranks of all t[n] in which #2's = #3's

A299242: ranks of all t[n] in which #2's < #3's

A299322: ranks of t[n] in which the 2's and 3's alternate

Rectangular arrays:

A299323: row n shows ranks of towers in which #2's = n

A299324: row n shows ranks of towers in which #3's = n

A299325: row n shows ranks of towers that start with n 2's

A299326: row n shows ranks of towers that start with n 3's

A299327: row n shows ranks of towers having maximal runlength n

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000

EXAMPLE

As an irregular triangle, where row n contains the digits of A248907(n):

2;

3;

2, 2;

2, 3;

3, 2;

2, 2, 2;

3, 3;

3, 2, 2;

2, 2, 3;

2, 3, 2;

3, 2, 3;

3, 3, 2;

2, 2, 2, 2;

3, 2, 2, 2;

2, 3, 3;

...

MATHEMATICA

t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};

t[6] = {2, 2, 2}; t[7] = {3, 3}; t[8] = {3, 2, 2}; t[9] = {2, 2, 3};

t[10] = {2, 3, 2}; t[11] = {3, 2, 3}; t[12] = {3, 3, 2};

z = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;

While[f < 13, n = f; While[n < z, p = 1;

While[p < 12, m = 2 n + 1; v = t[n]; k = 0;

While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];

p = p + 1; n = m]]; f = f + 1]

Flatten[Table[t[n], {n, 1, 120}]]; (* A299229 *)

CROSSREFS

KEYWORD

nonn,easy,tabf

AUTHOR

Clark Kimberling, Feb 06 2018

STATUS

approved