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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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-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.

LINKS

Table of n, a(n) for n=1..60.

EXAMPLE

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

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}]]  (* A299324, array *)

w[n_, k_] := r[n][[k]];

Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (* A299324, sequence *)

CROSSREFS

Cf. A299229, A299323.

Sequence in context: A035311 A182310 A244591 * A261076 A302991 A015791

Adjacent sequences:  A299321 A299322 A299323 * A299325 A299326 A299327

KEYWORD

nonn,easy,tabl

AUTHOR

Clark Kimberling, Feb 08 2018

STATUS

approved

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Last modified May 23 11:05 EDT 2019. Contains 323513 sequences. (Running on oeis4.)