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A299229
{2,3}-power towers in increasing order, concatenated; see Comments.
22
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, 3, 3, 3, 2, 3, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 3, 2, 2, 2, 2, 2
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
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 *)
KEYWORD
nonn,easy,tabf
AUTHOR
Clark Kimberling, Feb 06 2018
STATUS
approved