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A320158
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Decimal expansion of real root of x^3 + 11x^2 + 27x - 27 = 0, x^2*(x + 5)^3 - 2^2*3^3 = 0.
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3
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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
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0,1
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COMMENTS
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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).
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).
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LINKS
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EXAMPLE
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0.75310919791343792648241687233228285252957629792656...
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MAPLE
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evalf(solve(x^3+11*x^2+27*x-27=0, x)[1], 120); # Muniru A Asiru, Oct 07 2018
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MATHEMATICA
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Part[RealDigits[N[Root[x^3 + 11x^2 + 27x - 27 , 1], 100]], 1] (* Stefano Spezia, Oct 07 2018 *)
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PROG
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(PARI) default(realprecision, 100); solve(x=0, 1, x^3 + 11*x^2 + 27*x - 27)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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