A211666 - OEIS (original) (raw)

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COMMENTS

For a general definition like "Number of iterations log_p(log_p(log_p(...(n)...))) such that the result is < q", where p > 1, q > 0, the resulting g.f. is

g(x) = (1/(1-x))*Sum_{k>=1} x^(E_{i=1..k} b(i,k)), where b(i,k)=p for i<k and b(i,k)=q for i=k. The explicit first terms of the g.f. are g(x) = (x^q+x^(p^q)+x^(p^p^q)+x^(p^p^p^q)+...)/(1-x).

FORMULA

With the exponentiation definition E_{i=1..n} c(i) := c(1)^(c(2)^(c(3)^(...(c(n-1)^(c(n)))...))); E_{i=1..0} c := 1; example: E_{i=1..3} 10 = 10^(10^10) = 10^10000000000, we get:

a(E_{i=1..n} 10) = a(E_{i=1..n-1} 10)+1, for n>=1.

G.f.: g(x) = (1/(1-x))*Sum_{k>=1} x^(E_{i=1..k} b(i,k)), where b(i,k)=10 for i<k and b(i,k)=2 for i=k.

The explicit first terms of the g.f. are g(x) = (x^2+x^100+x^(10^100)+...)/(1-x).

EXAMPLE

a(n) = 0, 1, 2, 3 for n = 1, 2, 10^2, 10^10^2 (= 1, 2, 100, 10^100).