A359187 - OEIS

%I #30 Dec 10 2023 09:13:48

%S 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%T 0,0,0,0,0,0,0,0,0,0,0,4,9,9,4,1,7,5,2,6,5,5,0,1,4,7,5,0,5,2,6,9,1,3,

%U 4,6,3,9,1,3,5,4,1,2,8,6,7,6,7,4,5,4,0

%N Decimal expansion of the real part of (-sqrt(2))^^9, where ^^ indicates tetration or hyper-4 (e.g., 2^^4 = 2^(2^(2^2))).

%C This is the real part of (-sqrt(2))^(-sqrt(2))^(-sqrt(2))^(-sqrt(2))^(-sqrt(2))^(-sqrt(2))^(-sqrt(2))^(-sqrt(2))^(-sqrt(2)).

%C For a detailed explanation of the fact that a(2) = a(3) = a(4) = ... = a(45), see Links (i.e., the answer provided by the MathOverflow user Saúl RM).

%H Paolo Xausa, <a href="/A359187/b359187.txt">Table of n, a(n) for n = 1..10000</a>

%H MathOverflow, <a href="https://mathoverflow.net/questions/435085/the-9th-tetration-of-sqrt2/435088#435088">The 9th tetration of -sqrt(2)</a>

%H Thomas Scheuerle, <a href="/A359187/a359187.png">X: the tetration exponent of (-sqrt(2))^^X; Y: The phase angle of the complex result in units of Pi</a>.

%H Thomas Scheuerle, <a href="/A359187/a359187_1.png">X: the tetration exponent of (-sqrt(2))^^X; Y: The absolute value of the complex result</a>.

%e 1.000000000000000000000000000000000000000000004994175...

%p b:= n-> `if`(n=0, 1, (-sqrt(2))^b(n-1)):

%p evalf(Re(b(9)), 130);  # _Alois P. Heinz_, Nov 24 2023

%t First[RealDigits[Re[Nest[(-Sqrt[2])^#&,-Sqrt[2],8]],10,100]] (* _Paolo Xausa_, Oct 24 2023 *)

%o (PARI) localprec(100); real((-sqrt(2))^(-sqrt(2))^(-sqrt(2))^(-sqrt(2))^(-sqrt(2))^(-sqrt(2))^(-sqrt(2))^(-sqrt(2))^(-sqrt(2))) \\ _Michel Marcus_, Dec 20 2022

%Y Cf. A002193, A198094, A365937.

%K easy,cons,nonn

%O 1,46

%A _Marco Ripà_, Dec 19 2022