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A299323 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. 3
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, 57, 112, 111, 26, 22, 30, 39, 59, 113, 224, 223, 32, 23, 36, 43, 71, 115, 225, 448, 447, 33, 25, 37, 58, 79, 119, 227, 449, 896, 895, 50, 31, 40, 60, 87, 143, 231 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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..62.

EXAMPLE

Northwest corner:

   1     4     5    11    12    15

   3     8     9    10    18    20

   6    14    17    19    21    30

  13    28    29    35    39    43

  27    56    57    59    71    79

  55   112   113   115   119   143

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[#], 2] == n &]

TableForm[Table[r[n], {n, 1, 15}]]  (* A299323, array *)

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

Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A299323, sequence *)

CROSSREFS

Cf. A299229, A299324.

Sequence in context: A306835 A033546 A236360 * A010475 A256367 A242910

Adjacent sequences:  A299320 A299321 A299322 * A299324 A299325 A299326

KEYWORD

nonn,easy,tabl

AUTHOR

Clark Kimberling, Feb 08 2018

STATUS

approved

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Last modified May 23 07:46 EDT 2019. Contains 323508 sequences. (Running on oeis4.)