%I M3321 #57 Dec 11 2022 10:28:12

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

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

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

%N Decimal expansion of fifth root of 2.

%C The sine of 2017 times this number is the near-integer 0.999999999999999978567771261.... - _Alonso del Arte_, May 03 2013

%C With the present number r = 2^(1/5) and the golden section phi = A001622 the other (complex) roots of x^5 - 2 are given by x1 = r*exp(2*Pi*i/5) = r*(phi - 1 + sqrt(2 + phi)*i)/2 = r*(A001622 - 1 + A188593*i)/2 = 0.3549673131... + 1.0924770557...*i, x2 = r*exp(4*Pi*i/5) = r*(-phi + sqrt(3 - phi)*i)/2 = r*(-A001622 + A182007*i)/2 = -0.9293164906... + 0.6751879523...*i, and their complex conjugates. - _Wolfdieter Lang_, Dec 06 2022

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Harry J. Smith, <a href="/A005531/b005531.txt">Table of n, a(n) for n = 1..20000</a>

%F Equals Product_{k>=0} (1 + (-1)^k/(5*k + 4)). - _Amiram Eldar_, Jul 25 2020

%F From _Peter Bala_, Mar 02 2022: (Start)

%F Equals (3/2)*Sum_{n >= 0} (1/(5*n+2) - 1/(5*n-3))*binomial(1/5,n). Cf. A002580.

%F Equals (5/4)*hypergeom([-1/5, -3/5], [7/5], -1). (End)

%e 1.148698354997035006798626946777927589443850889097797505513711118493603... - _Harry J. Smith_, May 12 2009

%t RealDigits[N[2^(1/5),200]][[1]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 22 2012 *)

%o (PARI) { default(realprecision, 20080); x=2^(1/5); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b005531.txt", n, " ", d)); } \\ _Harry J. Smith_, May 12 2009

%Y Cf. A002950 (continued fraction).

%Y Cf. A002580 (cube root of 2).

%Y Cf. A001622, A182007, A188593.

%K nonn,cons

%O 1,3

%A _N. J. A. Sloane_

%E More terms from _Olaf Voß_, Feb 13 2008