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))); (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.

FORMULA

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)

EXAMPLE

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).

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 = 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 &]

CROSSREFS

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 07 2018

EXTENSIONS

a(37)-a(44) from Pontus von BrÃ¶mssen, Aug 08 2024

STATUS

approved