A320158 - OEIS

%I #25 Aug 21 2023 12:00:44

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

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

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

%N Decimal expansion of real root of x^3 + 11x^2 + 27x - 27 = 0, x^2*(x + 5)^3 - 2^2*3^3 = 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 = 3, so the upper bound is (x_0)/5 = 0.15062183958268758529648..., where x_0 = 0.75310919791343792648241... is the unique positive root to x^2*(x + 5)^3 - 2^2*3^3 = 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 = 0.72211990892483524148191... is the unique positive root of 2x^5 + 5x^2 - 3 = 0. For example, let p = 18053, q = 25000, t_1 = t_2 = ... = t_(18053^5) = -1, t_(18053^5+1) = t_(18053^5+2) = ... = t_(18053^5+25000^5) = 0.72212, then (Sum_{i=1..18053^5+25000^5} (t_i)^3)/(18053^5 + 25000^5) = 0.15062183958267256530376..., very close to (x_0)/5. Note that (x_0)/5 = ((t_0)^3 - (t_0)^5)/((t_0)^5 + 1).

%H Jianing Song, <a href="/A320158/b320158.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>

%e 0.75310919791343792648241687233228285252957629792656...

%p evalf(solve(x^3+11*x^2+27*x-27=0,x)[1],120); # _Muniru A Asiru_, Oct 07 2018

%t Part[RealDigits[N[Root[x^3 + 11x^2 + 27x - 27 ,1], 100]], 1] (* _Stefano Spezia_, Oct 07 2018 *)

%o (PARI) default(realprecision, 100); solve(x=0, 1, x^3 + 11*x^2 + 27*x - 27)

%Y Similar sequences: A320156 (k=5, m=1), A320157 (k=7, m=1).

%K nonn,cons

%O 0,1

%A _Jianing Song_, Oct 06 2018