

A299322


Ranks of {2,3}power towers without neither consecutive 2's nor consecutive 3's; see Comments.


1



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
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OFFSET

1,2


COMMENTS

Suppose that S is a set of real numbers. An Spowertower, 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(k1)); (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..36.


FORMULA

Conjectures from Colin Barker, Feb 09 2018: (Start)
G.f.: (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(n1) + a(n2)  a(n3) + 2*a(n4)  2*a(n5) for n>8.
(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

Cf. A299229.
Sequence in context: A109511 A018339 A128216 * A080735 A091856 A083416
Adjacent sequences: A299319 A299320 A299321 * A299323 A299324 A299325


KEYWORD

nonn,easy,new


AUTHOR

Clark Kimberling, Feb 07 2018


STATUS

approved



