|
| |
|
|
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
|
|
|
|
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
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
