OFFSET
0,1
COMMENTS
The first term of this sequence is given by floor(e[0]Pi) = floor(Pi + 1) = floor(4.14159) = 4, which is the integer part of "e zeration Pi". In general, zeration is not a commutative arithmetic operation, while floor(e[1]Pi) = floor(Pi + e) = floor(5.85987) = 5 and floor(e[2]Pi) = floor(Pi * e) = floor(8.53973) = 8 hold since e[1]Pi = Pi[1]e and e[2]Pi = Pi[2]e.
If n = 3, then floor(e[3]Pi) = floor(e^Pi) = floor(23.14069) = 23 (if n > 2, then hyper-n is not characterized by the commutative property anymore, even if we can find fascinating examples as 4[3]2 = 2[3]4 = 16).
Now, tetration can be extended to complex bases as described in the Paulsen reference and the corresponding term of the present sequence can be found using his online calculator (see Links), so we have that floor(e[4]Pi) = floor(37149801960.55) = 37149801960. An easy proof that 37149801960.55999 > e^^Pi > 37149801960.55 follows from the chain of inequalities 37149801960.5569855999 > |37149801960.5569855 + 5.9249049902894650649*10^(-11)| > e^^Pi > |37149801960.556985498 + 5.9249049902894650647*10^(-11)| > 37149801960.55.
As far as we know, it has not been proved if e^^Pi is an irrational number (or not).
LINKS
Hellmuth Kneser, Reelle analytische Lösungen der Gleichung phi(phi(x)) = e^x und verwandter Funktionalgleichungen, J. reine angew. Math. 187, 56-67 (1950)
Sheldon Levenstein (user sheldonison), New fatou.gp program, Jul 10 2015, updated Aug 14 2019.
William Paulsen, Tetration.
William Paulsen, Tetration for complex bases, Advances in Computational Mathematics, Vol. 45, No. 1 (2019), pp. 243-267; ResearchGate link.
Wikipedia, Hyperoperation
Wikipedia, Tetration
FORMULA
a(n) = floor(e[n]Pi).
EXAMPLE
For n = 3, a(3) = floor(e[3]Pi) = floor(e^Pi) = 15.
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
Marco Ripà, Apr 08 2022
STATUS
approved