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%I #4 Feb 08 2018 21:49:12
%S 1,4,3,5,8,6,11,9,14,13,12,10,17,28,27,15,18,19,29,56,55,24,20,21,35,
%T 57,112,111,26,22,30,39,59,113,224,223,32,23,36,43,71,115,225,448,447,
%U 33,25,37,58,79,119,227,449,896,895,50,31,40,60,87,143,231
%N Rectangular array by antidiagonals: row n gives the ranks of {2,3}-power towers in which the number of 2's is n; 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*. See A299229 for a guide to related sequences.
%e Northwest corner:
%e 1 4 5 11 12 15
%e 3 8 9 10 18 20
%e 6 14 17 19 21 30
%e 13 28 29 35 39 43
%e 27 56 57 59 71 79
%e 55 112 113 115 119 143
%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 t[8] = {3, 2, 2}; t[9] = {2, 2, 3}; t[10] = {2, 3, 2};
%t t[11] = {3, 2, 3}; t[12] = {3, 3, 2};
%t z = 400; g[k_] := If[EvenQ[k], {2}, {3}];
%t f = 6; While[f < 13, n = f; While[n < z, p = 1;
%t While[p < 18, 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 r[n_] := Select[Range[5000], Count[t[#], 2] == n &]
%t TableForm[Table[r[n], {n, 1, 15}]] (* A299323, array *)
%t w[n_, k_] := r[n][[k]];
%t Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A299323, sequence *)
%Y Cf. A299229, A299324.
%K nonn,easy,tabl
%O 1,2
%A _Clark Kimberling_, Feb 08 2018
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