A347177 - OEIS

A347177

Decimal expansion of real part of (i + (i + (i + (i + ...)^(1/3))^(1/3))^(1/3))^(1/3), where i is the imaginary unit.

%I #15 Apr 03 2024 05:07:17

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

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

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

%N Decimal expansion of real part of (i + (i + (i + (i + ...)^(1/3))^(1/3))^(1/3))^(1/3), where i is the imaginary unit.

%C This is the sum of the imaginary parts of the complex roots of the cubic equation 8*r^3 + 2*r - 1 = 0 , and its real solution is A347178. - _Gerry Martens_, Apr 02 2024

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NestedRadical.html">Nested Radical</a>

%F Equals cosh(asinh(3*sqrt(3)/2)/3). - _Gerry Martens_, Apr 02 2024

%e 1.1615413999972519360879176872471740748431...

%t RealDigits[(1/12) 2^(2/3) 3^(5/6) ((Sqrt[93] - 9)^(1/3) + (9 + Sqrt[93])^(1/3)), 10, 110][[1]]

%o (PARI) (1/12)*2^(2/3)*3^(5/6)*((sqrt(93) - 9)^(1/3) + (9 + sqrt(93))^(1/3)) \\ _Michel Marcus_, Aug 21 2021

%o (PARI) 2*imag(polroots(8*x^3 + 2*x - 1)[3]) \\ _Gerry Martens_, Apr 02 2024

%Y Cf. A060006, A156548, A156590, A347178.

%K nonn,cons

%O 1,3

%A _Ilya Gutkovskiy_, Aug 21 2021