A299229 -id:A299229 - cp's OEIS frontend

A299237 a(n) = the index m satisfying t(m) = reversal of t(n), where t(n) is the n-th {2,3}-power tower; see Comments.

1, 2, 3, 5, 4, 6, 7, 9, 8, 10, 11, 15, 13, 19, 12, 16, 21, 31, 14, 20, 17, 23, 22, 32, 25, 33, 27, 39, 43, 63, 18, 24, 26, 34, 35, 47, 51, 67, 28, 40, 44, 64, 29, 41, 45, 65, 36, 48, 52, 68, 37, 49, 53, 69, 55, 79, 87, 127, 71, 95, 103, 135, 30, 42, 46, 66

Offset: 1

Clark Kimberling, Feb 07 2018

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.

This sequence is a self-inverse permutation of the positive integers.

			t(12) = (3,3,2) and t(15) = (2,3,3) = reversal of t(12); therefore a(12) = 15.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A299229, A299239 (fixed points of the permutation; palindromes).

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 = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;
While[f < 13, n = f; While[n < z, p = 1;
  While[p < 12, 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_] := Reverse[t[n]]
Flatten[Table[Select[Range[2000], t[#] == r[n] &], {n, 1, 1500}]]; (* A299237 *)

A299239 Ranks of palindromic {2,3}-power towers; see Comments.

Original entry on oeis.org

1, 2, 3, 6, 7, 10, 11, 13, 16, 20, 25, 27, 34, 35, 40, 45, 48, 53, 55, 66, 70, 75, 80, 89, 100, 109, 111, 119, 130, 142, 147, 155, 160, 168, 177, 185, 196, 204, 213, 221, 223, 247, 258, 266, 278, 286, 291, 315, 320, 344, 353, 377, 388, 412, 421, 445, 447, 463

Offset: 1

Clark Kimberling, Feb 07 2018

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.

			The first six palindromes are t(1) = (2), t(2) = (3), t(3) = (2,2), t(6) = (2,2,2), t(7) = (3,3), t(10) = (2,3,2).

Pontus von Brömssen, Table of n, a(n) for n = 1..10000 (terms 1..280 from Clark Kimberling)

Cf. A299229, A299237.

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 = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;
While[f < 13, n = f; While[n < z, p = 1;
  While[p < 12, 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]
Flatten[Table[Select[Range[1000], t[#] == Reverse[t[#]] &], {n, 1, 120}]]

A299230 a(n) = height of n-th {2,3}-power tower; see Comments.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 4, 4, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Clark Kimberling, Feb 06 2018

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.

			t(8) = (3,2,2), so that a(8) = 3.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A299229.

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 = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;
While[f < 13, n = f; While[n < z, p = 1;
  While[p < 12, 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]
Flatten[Table[t[n], {n, 1, 120}]];    (* A299229 *)
w = Table[Length[t[n]], {n, 1, 120}]; (* A299230 *)

A299238 a(n) = the index m satisfying t(m) = 5 - t(n), where t(n) is the n-th {2,3}-power tower; see Comments.

Original entry on oeis.org

2, 1, 7, 5, 4, 16, 3, 15, 12, 11, 10, 9, 34, 33, 8, 6, 32, 31, 26, 25, 24, 23, 22, 21, 20, 19, 70, 69, 68, 67, 18, 17, 14, 13, 66, 65, 64, 63, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 142, 141, 140, 139, 138, 137, 136, 135, 38, 37, 36

Offset: 1

Clark Kimberling, Feb 07 2018

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.

This sequence is a self-inverse permutation of the positive integers.

			t(12) = (3,3,2) and t(9) = (2,2,3) = 5 - (3,3,2), so that a(12) = 9. (Note: 5 - (x(1), x(2), ..., x(k)) means (5-x(1), 5-x(2), ..., 5-x(k))).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A299229.

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 = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;
While[f < 13, n = f; While[n < z, p = 1;
  While[p < 12, 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]
Flatten[Table[Select[Range[1000], t[#] == 5 - t[n] &], {n, 1, 150}]] (* A299238 *)

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.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Feb 08 2018

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.

			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

Cf. A299229, A299324.

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}]]  (* this array *)
w[n_, k_] := r[n][[k]];
Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* this sequence *)

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.

Original entry on oeis.org

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

Clark Kimberling, Feb 08 2018

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.

			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

Cf. A299229, A299323.

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 *)

A299325 Rectangular array by antidiagonals: row n gives the ranks of {2,3}-power towers that start with n 2's, for n >= 1; see Comments.

Original entry on oeis.org

1, 4, 3, 10, 9, 6, 15, 21, 19, 13, 17, 31, 43, 39, 27, 23, 35, 63, 87, 79, 55, 25, 47, 71, 127, 175, 159, 111, 29, 51, 95, 143, 255, 351, 319, 223, 33, 59, 103, 191, 287, 511, 703, 639, 447, 37, 67, 119, 207, 383, 575, 1023, 1407, 1279, 895, 41, 75, 135, 239

Offset: 1

Clark Kimberling, Feb 08 2018

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.

As sequences, this one and A299326 partition the positive integers.

			Northwest corner:
   1    4    10    15    17    23    25
   3    9    21    31    35    47    51
   6   19    43    63    71    95   103
  13   39    87   127   143   191   207
  27   79   175   255   287   383   415

Cf. A299229, A299326.

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 = 500; g[k_] := If[EvenQ[k], {2}, {3}];
f = 6; While[f < 13, n = f;  While[n < z, p = 1;
   While[p < 17, 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]
s = Select[Range[60000], Count[First[Split[t[#]]], 3] == 0 & ];
r[n_] := Select[s, Length[First[Split[t[#]]]] == n &, 12]
TableForm[Table[r[n], {n, 1, 11}]]  (* this array *)
w[n_, k_] := r[n][[k]];
Table[w[n - k + 1, k], {n, 11}, {k, n, 1, -1}] // Flatten (* this sequence *)

A299326 Rectangular array by antidiagonals: row n gives the ranks of {2,3}-power towers that start with n 3's, for n >= 1; see Comments.

Original entry on oeis.org

2, 5, 7, 8, 12, 16, 11, 18, 26, 34, 14, 24, 38, 54, 70, 20, 30, 50, 78, 110, 142, 22, 42, 62, 102, 158, 222, 286, 28, 46, 86, 126, 206, 318, 446, 574, 32, 58, 94, 174, 254, 414, 638, 894, 1150, 36, 66, 118, 190, 350, 510, 830, 1278, 1790, 2302

Offset: 1

Clark Kimberling, Feb 08 2018

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.

As sequences, this one and A299325 partition the positive integers.

			Northwest corner:
2     5    8   11   14   20   22
7    12   18   24   30   42   46
16   26   38   50   62   86   94
34   54   78  102  126  174  190
70  110  158  206  254  350  382

1

Cf. A299229, A299325.

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 = 500; g[k_] := If[EvenQ[k], {2}, {3}];
f = 6; While[f < 13, n = f; While[n < z, p = 1;
   While[p < 17, 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]
s = Select[Range[60000], Count[First[Split[t[#]]], 2] == 0 & ];
r[n_] := Select[s, Length[First[Split[t[#]]]] == n &, 12]
TableForm[Table[r[n], {n, 1, 10}]]  (* this array *)
w[n_, k_] := r[n][[k]];
Table[w[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* this sequence *)

A375376 Square array read by antidiagonals: Let n = Sum_{i=1..m} 2^e_i be the binary expansion of n, let S be the set {e_i+2; 1 <= i <= m}, and let X be the sequence of power towers built of numbers in S, sorted first by their height and then colexicographically. The n-th row of the array gives the permutation of indices which reorders X by magnitude. In case of ties, keep the colexicographic order.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 6, 5, 5, 3, 2, 1, 7, 6, 4, 4, 3, 2, 1, 8, 7, 7, 5, 4, 3, 2, 1, 9, 8, 6, 6, 5, 4, 3, 2, 1, 10, 9, 8, 7, 7, 5, 4, 3, 3, 1, 11, 10, 11, 8, 6, 6, 7, 4, 2, 2, 1, 12, 11, 9, 9, 8, 7, 5, 5, 7, 3, 2, 1, 13, 12, 12, 10, 9, 8, 6, 6, 4, 4, 4, 2, 1

Offset: 1

Pontus von Brömssen, Aug 14 2024

Each row is a permutation of the positive integers.

If n is a power of 2, the set S contains a single number and the n-th row is the identity permutation.

			Array begins:
   n=1: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=2: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=3: 1, 2, 3, 5, 4, 7,  6,  8, 11,  9, 12, 10, 15, 16, 13, ...
   n=4: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=5: 1, 2, 3, 4, 5, 7,  6,  8,  9, 11, 15, 10, 12, 16, 13, ...
   n=6: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=7: 1, 2, 3, 4, 7, 5,  6, 10, 13,  8,  9, 11, 14, 12, 15, ...
   n=8: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=9: 1, 3, 2, 7, 4, 5,  8,  6, 15,  9, 11, 16, 10, 12, 17, ...
  n=10: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
  n=11: 1, 2, 4, 3, 7, 5, 13,  6,  8, 10, 14,  9, 11, 22, 16, ...
  n=12: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
  n=13: 1, 2, 4, 3, 5, 7, 13,  6, 10,  8, 14,  9, 15, 11, 12, ...
  n=14: 1, 2, 3, 4, 5, 7,  6, 10,  8,  9, 11, 12, 13, 14, 15, ...
  n=15: 1, 2, 3, 5, 4, 9,  6,  7, 13, 21,  8, 10, 17, 11, 14, ...
For n = 7 = 2^0 + 2^1 + 2^2, the set S is {0+2, 1+2, 2+2} = {2, 3, 4}. The smallest power towers formed by 2's, 3's, and 4's, together with their colex ranks are:
   k | power tower | colex rank T(7,k)
   --+-------------+------------------
   1 |     2 = 2   |        1
   2 |     3 = 3   |        2
   3 |     4 = 4   |        3
   4 |   2^2 = 4   |        4
   5 |   2^3 = 8   |        7
   6 |   3^2 = 9   |        5
   7 |   4^2 = 16  |        6
   8 |   2^4 = 16  |       10
   9 | 2^2^2 = 16  |       13
  10 |   3^3 = 27  |        8
  11 |   4^3 = 64  |        9
  12 |   3^4 = 81  |       11
  13 | 3^2^2 = 81  |       14
  14 |   4^4 = 256 |       12
  15 | 4^2^2 = 256 |       15

Index entries for sequences that are permutations of the natural numbers.

Cf. A185969, A299229, A375374 (3rd row), A375377 (the inverse permutation to each row).

A299322 Ranks of {2,3}-power towers with neither consecutive 2's nor consecutive 3's; see Comments.

Original entry on oeis.org

1, 2, 4, 5, 10, 11, 22, 23, 45, 48, 92, 97, 185, 196, 372, 393, 745, 788, 1492, 1577, 2985, 3156, 5972, 6313, 11945, 12628, 23892, 25257, 47785, 50516, 95572, 101033, 191145, 202068, 382292, 404137, 764585, 808276, 1529172, 1616553, 3058345, 3233108, 6116692, 6466217

Offset: 1

Clark Kimberling, Feb 07 2018

nonn

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.

			The first seven terms are the ranks of these towers: t(1) = (2), t(2) = (3), t(4) = (2,3), t(5) = (3,2), t(10) = (2,3,2), t(11) = (3,2,3), t(22) = (3,2,3,2).

Cf. A299229.

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 = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;
While[f < 13, n = f; While[n < z, p = 1;
  While[p < 12, 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]
Select[Range[5000], Max[Map[Length, Split[t[#]]]] < 2 &]

Conjectures from Colin Barker, Feb 09 2018: (Start)
G.f.: x*(1 + x + x^2 + x^4 - 2*x^5 + 2*x^6 - 2*x^7 + x^8) / ((1 - x)*(1 + x^2)*(1 - 2*x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) + 2*a(n-4) - 2*a(n-5) for n >= 10.
(End)

a(37)-a(44) from Pontus von Brömssen, Aug 08 2024

cp's OEIS Frontend

A299237 a(n) = the index m satisfying t(m) = reversal of t(n), where t(n) is the n-th {2,3}-power tower; see Comments.

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A299239 Ranks of palindromic {2,3}-power towers; see Comments.

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A299230 a(n) = height of n-th {2,3}-power tower; see Comments.

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A299238 a(n) = the index m satisfying t(m) = 5 - t(n), where t(n) is the n-th {2,3}-power tower; see Comments.

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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.

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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.

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A299325 Rectangular array by antidiagonals: row n gives the ranks of {2,3}-power towers that start with n 2's, for n >= 1; see Comments.

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A299326 Rectangular array by antidiagonals: row n gives the ranks of {2,3}-power towers that start with n 3's, for n >= 1; see Comments.

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