A299324 Rectangular array by antidiagonals: row n gives the ranks of {2,3}-power towers in which the number of 3's is n; see Comments.

2, 4, 7, 5, 11, 16, 8, 12, 24, 34, 9, 15, 26, 50, 70, 10, 18, 32, 54, 102, 142, 14, 20, 33, 66, 110, 206, 286, 17, 22, 38, 68, 134, 222, 414, 574, 19, 23, 42, 69, 138, 270, 446, 830, 1150, 21, 25, 46, 78, 140, 278, 542, 894, 1662, 2302, 28, 30, 48, 86, 141

Offset: 1

Author

Clark Kimberling, Feb 08 2018

Keywords

• nonn
• easy
• tabl

Comments

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.

Examples

			Northwest corner:
   2     4    5     8     9    10
   7    11   12    15    18    20
  16    24   26    32    33    38
  34    50   54    66    68    69
  70   102   110  134   138   140
		

Crossrefs

Cf. A299229, A299323.

Programs

Mathematica

t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};
t[6] = {2, 2, 2}; t[7] = {3, 3};
t[8] = {3, 2, 2}; t[9] = {2, 2, 3}; t[10] = {2, 3, 2};
t[11] = {3, 2, 3}; t[12] = {3, 3, 2};
z = 400; g[k_] := If[EvenQ[k], {2}, {3}];
f = 6; While[f < 13, n = f; While[n < z, p = 1;
   While[p < 18, m = 2 n + 1; v = t[n]; k = 0;
   While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];
   p = p + 1; n = m]]; f = f + 1]
r[n_] := Select[Range[5000], Count[t[#], 3] == n &]
TableForm[Table[r[n], {n, 1, 15}]]  (* this array *)
w[n_, k_] := r[n][[k]];
Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (* this sequence *)