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A320156 Decimal expansion of the unique real root of x^3 - 3*x^2 + 8*x - 16 = 0, or equivalently, the unique positive root of x^4*(x + 5) - 4^4 = 0. 3

%I

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

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

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

%N Decimal expansion of the unique real root of x^3 - 3*x^2 + 8*x - 16 = 0, or equivalently, the unique positive root of x^4*(x + 5) - 4^4 = 0.

%C Let t_1, t_2, ..., t_n be n real numbers in [-1, 1] such that Sum_{i=1..n} (t_i)^k = 0, then lim sup ((Sum_{i=1..n} (t_i)^m)/n) = (x_0)/k, where k > m > 0 are odd numbers and x_0 is the unique positive root of x^(k-m)*(x + k)^m - m^m*(k - m)^(k-m). x_0 is an algebraic integer of degree k - 2. The upper bound (x_0)/k can be approached by x_1 = x_2 = ... = x_(p^k) = -1, x_(p^k+1) = x_(p^k+2) = ... = x_(p^k+q^k) = p/q, where p/q is a rational arbitrarily close to t_0, t_0 is the unique positive root of (k - m)*x^k + k*x^(k - m) - m = 0. Note that (x_0)/k = ((t_0)^m - (t_0)^k)/((t_0)^k + 1).

%C Here k = 5, m = 1, so the upper bound is (x_0)/5 = 0.48466366895061441679275..., where x_0 = 2.4233183447530720839637... is the unique positive root to x^4*(x + 5) - 4^4 = 0. The upper bound (x_0)/5 can be approached by t_1 = t_2 = ... = t_(p^5) = -1, t_(p^5+1) = t_(p^5+2) = ... = t_(p^5+q^5) = p/q, where p/q is a rational arbitrarily close to t_0, t_0 = 0.60582958618826802099093... is the unique positive root of 4*x^5 + 5*x^4 - 1 = 0. For example, let p = 60583, q = 100000, t_1 = t_2 = ... = t_(60583^5) = -1, t_(60583^5+1) = t_(60583^5+2) = ... = t_(60583^5+100000^5) = 0.60583, then (Sum_{i=1..60583^5+100000^5} t_i)/(60583^5 + 100000^5) = 0.48466366895009176321695..., very close to (x_0)/5. Note that (x_0)/5 = (t_0 - (t_0)^5)/((t_0)^5 + 1).

%H Jianing Song, <a href="/A320156/b320156.txt">Table of n, a(n) for n = 1..10000</a>

%e 2.4233183447530720839637549246282910396018770662662...

%p evalf(solve(x^3-3*x^2+8*x-16=0,x)[1],120); # _Muniru A Asiru_, Oct 07 2018

%t Part[RealDigits[N[Root[x^3 - 3x^2 + 8x - 16 ,1], 100]], 1] (* _Stefano Spezia_, Oct 07 2018 *)

%o (PARI) default(realprecision, 100); solve(x=2, 3, x^3 - 3*x^2 + 8*x - 16)

%o (Sage) (x^3-3*x^2+8*x-16==0).find_root(2,3,x) # _G. C. Greubel_, Feb 25 2019

%Y Similar sequences: A320157 (k=7, m=1), A320158 (k=5, m=3).

%K nonn,cons

%O 1,1

%A _Jianing Song_, Oct 06 2018

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Last modified November 18 12:14 EST 2019. Contains 329261 sequences. (Running on oeis4.)