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A299327 Rectangular array by antidiagonals: row n gives the ranks of {2,3}-power towers in which the maximal runlength is n; see Comments.

%I A299327 #15 Aug 08 2024 08:31:57
%S A299327 1,2,3,4,7,6,5,8,14,13,10,9,16,28,27,11,12,19,34,56,55,22,15,26,39,70,
%T A299327 112,111,23,17,29,54,79,142,224,223,45,18,30,57,110,159,286,448,447,
%U A299327 48,20,33,58,113,222,319,574,896,895,92,21,38,69,114,225,446
%N A299327 Rectangular array by antidiagonals: row n gives the ranks of {2,3}-power towers in which the maximal runlength is n; see Comments.
%C A299327 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.
%e A299327 Northwest corner:
%e A299327    1   2   4   5   10   11   22   23   45   48
%e A299327    3   7   8   9   12   15   17   18   20   21
%e A299327    6  14  16  19   26   29   30   33   38   40
%e A299327   13  28  34  39   54   57   58   69   78   80
%e A299327   27  56  70  79  110  113  114  141  158  160
%t A299327 t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};
%t A299327 t[6] = {2, 2, 2}; t[7] = {3, 3};
%t A299327 t[8] = {3, 2, 2}; t[9] = {2, 2, 3}; t[10] = {2, 3, 2};
%t A299327 t[11] = {3, 2, 3}; t[12] = {3, 3, 2};
%t A299327 z = 500; g[k_] := If[EvenQ[k], {2}, {3}];
%t A299327 f = 6; While[f < 13, n = f; While[n < z, p = 1;
%t A299327 While[p < 15, m = 2 n + 1; v = t[n]; k = 0;
%t A299327 While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];
%t A299327 p = p + 1; n = m]]; f = f + 1]
%t A299327 r[n_] := Select[Range[30000], Max[Map[Length, Split[t[#]]]] == n & , 12];
%t A299327 TableForm[Table[r[n], {n, 1, 12}]]  (* this array *)
%t A299327 w[n_, k_] := r[n][[k]];
%t A299327 Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* this sequence *)
%Y A299327 Cf. A299229.
%K A299327 nonn,easy,tabl
%O A299327 1,2
%A A299327 _Clark Kimberling_, Feb 08 2018