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A299322 Ranks of {2,3}-power towers with neither consecutive 2's nor consecutive 3's; see Comments.

%I A299322 #23 Aug 09 2024 11:50:26
%S A299322 1,2,4,5,10,11,22,23,45,48,92,97,185,196,372,393,745,788,1492,1577,
%T A299322 2985,3156,5972,6313,11945,12628,23892,25257,47785,50516,95572,101033,
%U A299322 191145,202068,382292,404137,764585,808276,1529172,1616553,3058345,3233108,6116692,6466217
%N A299322 Ranks of {2,3}-power towers with neither consecutive 2's nor consecutive 3's; see Comments.
%C A299322 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*. See A299229 for a guide to related sequences.
%F A299322 Conjectures from _Colin Barker_, Feb 09 2018: (Start)
%F A299322 G.f.: x*(1 + x + x^2 + x^4 - 2*x^5 + 2*x^6 - 2*x^7 + x^8) / ((1 - x)*(1 + x^2)*(1 - 2*x^2)).
%F A299322 a(n) = a(n-1) + a(n-2) - a(n-3) + 2*a(n-4) - 2*a(n-5) for n >= 10.
%F A299322 (End)
%e A299322 The first seven terms are the ranks of these towers: t(1) = (2), t(2) = (3), t(4) = (2,3), t(5) = (3,2), t(10) = (2,3,2), t(11) = (3,2,3), t(22) = (3,2,3,2).
%t A299322 t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};
%t A299322 t[6] = {2, 2, 2}; t[7] = {3, 3}; t[8] = {3, 2, 2}; t[9] = {2, 2, 3};
%t A299322 t[10] = {2, 3, 2}; t[11] = {3, 2, 3}; t[12] = {3, 3, 2};
%t A299322 z = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;
%t A299322 While[f < 13, n = f; While[n < z, p = 1;
%t A299322   While[p < 12, m = 2 n + 1; v = t[n]; k = 0;
%t A299322     While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];
%t A299322    p = p + 1; n = m]]; f = f + 1]
%t A299322 Select[Range[5000], Max[Map[Length, Split[t[#]]]] < 2 &]
%Y A299322 Cf. A299229.
%K A299322 nonn
%O A299322 1,2
%A A299322 _Clark Kimberling_, Feb 07 2018
%E A299322 a(37)-a(44) from _Pontus von Brömssen_, Aug 08 2024