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%I #11 Jan 18 2022 03:36:14
%S 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,
%T 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,
%U 2,3,3,3,2,3,2,3,3,2,3,3,3,2,2,2,2,2
%N {2,3}-power towers in increasing order, concatenated; see Comments.
%C 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-1)); (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*.
%C 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)).
%C A299229: sequence of all {2,3}-towers, ranked, concatenated
%C A299230: a(n) = height of t(n)
%C A299231: all n such that t(n) has x(1) = 2
%C A299232: all n such that t(n) has x(1) = 3
%C A299233: all n such that t(n) has x(k) = 2
%C A299234: all n such that t(n) has x(k) = 3
%C A299235: a(n) = number of 2's in t(n)
%C A299236: a(n) = number of 3's in t(n)
%C A299237: a(n) = m satisfying t(m) = reversal of t(n)
%C A299238; a(n) = m satisfying t(m) = 5 - t(n)
%C A999239: all n such that t(n) is a palindrome
%C A229240: ranks of all t[n] in which #2's > #3's
%C A299241: ranks of all t[n] in which #2's = #3's
%C A299242: ranks of all t[n] in which #2's < #3's
%C A299322: ranks of t[n] in which the 2's and 3's alternate
%C Rectangular arrays:
%C A299323: row n shows ranks of towers in which #2's = n
%C A299324: row n shows ranks of towers in which #3's = n
%C 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
%C A299327: row n shows ranks of towers having maximal runlength n
%H Clark Kimberling, <a href="/A299229/b299229.txt">Table of n, a(n) for n = 1..10000</a>
%e The first twelve {2,3}-power towers, ranked:
%e t(1) = (2)
%e t(2) = (3)
%e t(3) = (2,2)
%e t(4) = (2,3)
%e t(5) = (3,2)
%e t(6) = (2,2,2)
%e t(7) = (3,3)
%e t(8) = (3,2,2)
%e t(9) = (2,2,3)
%e t(10) = (2,3,2)
%e t(11) = (2,3,2)
%e t(12) = (3,3,2)
%e Concatening these towers gives the first 28 terms of the sequence.
%t t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};
%t t[6] = {2, 2, 2}; t[7] = {3, 3}; t[8] = {3, 2, 2}; t[9] = {2, 2, 3};
%t t[10] = {2, 3, 2}; t[11] = {3, 2, 3}; t[12] = {3, 3, 2};
%t z = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;
%t While[f < 13, n = f; While[n < z, p = 1;
%t While[p < 12, m = 2 n + 1; v = t[n]; k = 0;
%t While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];
%t p = p + 1; n = m]]; f = f + 1]
%t Flatten[Table[t[n], {n, 1, 120}]]; (* A299229 *)
%Y Cf. A299230-A229242, A256231, A185969, A299322-A299327.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, Feb 06 2018
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