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%I #15 Jul 25 2021 20:41:15
%S 5,9,6,0,7,1,6,3,7,9,8,3,3,2,1,5,2,3,1,1,2,8,0,5,4,1,4,3,9,9,6,8,1,8,
%T 2,8,1,1,3,3,2,5,4,9,4,3,9,6,2,1,3,1,9,4,3,2,4,7,9,0,8,3,0,3,6,0,0,5,
%U 1,6,2,6,8,6,2,0,8,9,1,8,5,8,7,1,6,6,0,3,3,7,5,4,2,8,4,7,5,4,7,3
%N Decimal expansion of the real solution to equation x^3 + 3*x = 2.
%C A014176 Decimal expansion of the silver mean, 1+sqrt(2). Interestingly, when we look for x=0.596071637983321523112805414399681828113325, the original Inverse Symbolic Calculator replies: "... Roots of polynomials of 4th degree (coeffs: -9..9) 5960716379833215 = 10+15*x+5*x^3" - with wrong sign of the first term.
%C Let p(n,x) denote the n-th Maclaurin polynomial of e^x, and let p'(n,x) denote its derivative. Then p'(n+1,x) = p(n,x), so that the real zero r of p(n,x), for odd n, is also the value of x that minimizes p(n+1,x). Let y = 0.5960716... . Then for n = 3, we have r = - y - 1; see A332324 for the minimal value of p(4,x). - _Clark Kimberling_, Feb 13 2020
%H <a href="http://wayback.cecm.sfu.ca/projects/ISC/ISCmain.html">Original Inverse Symbolic Calculator</a>
%F x = (1+sqrt(2))^(1/3) - 1/(1+sqrt(2))^(1/3).
%e x = 0.596071637983321523112805414399681828113325...
%t RealDigits[N[Solve[3*z+z^3==2,z][[1,1,2]],100]][[1]]
%Y Cf. A014176, A332324.
%K cons,nonn
%O 0,1
%A _Zak Seidov_, Apr 27 2006
%E Website URL and name (changed from "Plouffe's Inverter") updated by _Jon E. Schoenfield_, Feb 28 2020
%E a(99) corrected by _Sean A. Irvine_, Jul 25 2021
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