

A320158


Decimal expansion of real root of x^3 + 11x^2 + 27x  27 = 0, x^2*(x + 5)^3  2^2*3^3 = 0.


3



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, 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, 0, 7, 6, 5, 4, 2, 8, 7, 3, 6, 0, 0, 8, 9, 7, 7, 2, 4, 1
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OFFSET

0,1


COMMENTS

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^(km)*(x + k)^m  m^m*(k  m)^(km). 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).
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).


LINKS

Jianing Song, Table of n, a(n) for n = 0..10000


EXAMPLE

0.75310919791343792648241687233228285252957629792656...


MAPLE

evalf(solve(x^3+11*x^2+27*x27=0, x)[1], 120); # Muniru A Asiru, Oct 07 2018


MATHEMATICA

Part[RealDigits[N[Root[x^3 + 11x^2 + 27x  27 , 1], 100]], 1] (* Stefano Spezia, Oct 07 2018 *)


PROG

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


CROSSREFS

Similar sequences: A320156 (k=5, m=1), A320157 (k=7, m=1).
Sequence in context: A245510 A199794 A106040 * A085927 A180597 A219242
Adjacent sequences: A320155 A320156 A320157 * A320159 A320160 A320161


KEYWORD

nonn,cons


AUTHOR

Jianing Song, Oct 06 2018


STATUS

approved



