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A299326 Rectangular array by antidiagonals: row n gives the ranks of {2,3}-power towers that start with n 3's, for n >= 1; see Comments.

%I A299326 #14 Aug 07 2024 15:21:55
%S A299326 2,5,7,8,12,16,11,18,26,34,14,24,38,54,70,20,30,50,78,110,142,22,42,
%T A299326 62,102,158,222,286,28,46,86,126,206,318,446,574,32,58,94,174,254,414,
%U A299326 638,894,1150,36,66,118,190,350,510,830,1278,1790,2302
%N A299326 Rectangular array by antidiagonals: row n gives the ranks of {2,3}-power towers that start with n 3's, for n >= 1; see Comments.
%C A299326 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.
%C A299326 As sequences, this one and A299325 partition the positive integers.
%D A299326 1
%e A299326 Northwest corner:
%e A299326 2     5    8   11   14   20   22
%e A299326 7    12   18   24   30   42   46
%e A299326 16   26   38   50   62   86   94
%e A299326 34   54   78  102  126  174  190
%e A299326 70  110  158  206  254  350  382
%t A299326 t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};
%t A299326 t[6] = {2, 2, 2}; t[7] = {3, 3};
%t A299326 t[8] = {3, 2, 2}; t[9] = {2, 2, 3}; t[10] = {2, 3, 2};
%t A299326 t[11] = {3, 2, 3}; t[12] = {3, 3, 2};
%t A299326 z = 500; g[k_] := If[EvenQ[k], {2}, {3}];
%t A299326 f = 6; While[f < 13, n = f; While[n < z, p = 1;
%t A299326    While[p < 17, m = 2 n + 1; v = t[n]; k = 0;
%t A299326    While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];
%t A299326    p = p + 1; n = m]]; f = f + 1]
%t A299326 s = Select[Range[60000], Count[First[Split[t[#]]], 2] == 0 & ];
%t A299326 r[n_] := Select[s, Length[First[Split[t[#]]]] == n &, 12]
%t A299326 TableForm[Table[r[n], {n, 1, 10}]]  (* this array *)
%t A299326 w[n_, k_] := r[n][[k]];
%t A299326 Table[w[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* this sequence *)
%Y A299326 Cf. A299229, A299325.
%K A299326 nonn,easy,tabl
%O A299326 1,1
%A A299326 _Clark Kimberling_, Feb 08 2018