

A299229


{2,3}power towers in increasing order, concatenated; see Comments.


20



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
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OFFSET

1,1


COMMENTS

Suppose that S is a set of real numbers. An Spowertower, 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(k1)); (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 "powertower", and t(n) denotes the nth {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)
A999239: all n such that t(n) is a palindrome
A229240: 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'sA299326: 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

The first twelve {2,3}power towers, ranked:
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) = (2,3,2)
t(12) = (3,3,2)
Concatening these towers gives the first 28 terms of the sequence.


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

Cf. A299230A229242, A256231, A185969, A299322A299327.
Sequence in context: A210659 A103266 A185150 * A289496 A248973 A305048
Adjacent sequences: A299226 A299227 A299228 * A299230 A299231 A299232


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Feb 06 2018


STATUS

approved



