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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.

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`%I #13 Aug 07 2024 14:24:08
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`%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,
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`%T 57,112,111,26,22,30,39,59,113,224,223,32,23,36,43,71,115,225,448,447,
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`%U 33,25,37,58,79,119,227,449,896,895,50,31,40,60,87,143,231
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`%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.
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`%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))); (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.
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`%e Northwest corner:
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`%e 1 4 5 11 12 15
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`%e 3 8 9 10 18 20
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`%e 6 14 17 19 21 30
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`%e 13 28 29 35 39 43
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`%e 27 56 57 59 71 79
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`%e 55 112 113 115 119 143
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`%t t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};
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`%t t[6] = {2, 2, 2}; t[7] = {3, 3};
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`%t t[8] = {3, 2, 2}; t[9] = {2, 2, 3}; t[10] = {2, 3, 2};
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`%t t[11] = {3, 2, 3}; t[12] = {3, 3, 2};
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`%t z = 400; g[k_] := If[EvenQ[k], {2}, {3}];
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`%t f = 6; While[f < 13, n = f; While[n < z, p = 1;
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`%t While[p < 18, m = 2 n + 1; v = t[n]; k = 0;
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`%t While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];
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`%t p = p + 1; n = m]]; f = f + 1]
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`%t r[n_] := Select[Range[5000], Count[t[#], 2] == n &]
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`%t TableForm[Table[r[n], {n, 1, 15}]] (* this array *)
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`%t w[n_, k_] := r[n][[k]];
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`%t Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* this sequence *)
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`%Y Cf. A299229, A299324.
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`%K nonn,easy,tabl
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`%O 1,2
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`%A _Clark Kimberling_, Feb 08 2018
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