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%I #51 Aug 14 2023 10:40:39
%S 7,5,4,8,7,7,6,6,6,2,4,6,6,9,2,7,6,0,0,4,9,5,0,8,8,9,6,3,5,8,5,2,8,6,
%T 9,1,8,9,4,6,0,6,6,1,7,7,7,2,7,9,3,1,4,3,9,8,9,2,8,3,9,7,0,6,4,6,0,8,
%U 0,6,5,5,1,2,8,0,8,1,0,9,0,7,3,8,2,2,7,0,9,2,8,4,2,2,5,0,3,0,3,6,4,8,3,7,7
%N Decimal expansion of the real root of x^3 + x^2 - 1.
%C Also decimal expansion of the root of x^(1/sqrt(x+1)) = (1/sqrt(x+1))^x. The root of (1/x)^(1/sqrt(x+1)) = (1/sqrt(x+1))^(1/x) is the golden ratio. - _Michel Lagneau_, Apr 17 2012
%C The following decomposition holds true: X^3 + X^2 - 1 = (X - r)*(X + i * e^(-i*a) * r^(-1/2))*(X - i * e^(i*a) * r^(-1/2)), where a = arcsin(1/(2*r^(3/2))), see A218197 for the decimal expansion of a and the paper of Witula et al. for details. - _Roman Witula_, Oct 22 2012
%D Roman Witula, E. Hetmaniok, D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, submitted to Proceedings of the 15th International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.
%H H. R. P. Ferguson, <a href="http://www.fq.math.ca/Scanned/14-3/ferguson.pdf">On a Generalization of the Fibonacci Numbers Useful in Memory Allocation Schema or All About the Zeroes of Z^k - Z^{k - 1} - 1, k > 0</a>, Fibonacci Quarterly, Volume 14, Number 3, October, 1976 (see Table 2 p. 238).
%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%F Let 0 < a < 1 be any real number. Then a is the lesser and 1 is the greater and a^2/1 = 1/(a+1) and a^3 + a^2 - 1 = 0. Solving this using PARI we have 0.7548776662466927600495088964... . The general cubic can also be solved in radicals.
%F Equals -(1/3) + (1/3)*(25/2 - (3*sqrt(69))/2)^(1/3) + (1/3)*((1/2)*(25 + 3*sqrt(69)))^(1/3).
%e 0.7548776662466927600495088963585286918946066...
%p A075778 := proc()
%p 1/3-root[3](25/2-3*sqrt(69)/2)/3 -root[3](25/2+3*sqrt(69)/2)/3;
%p -% ;
%p end proc: # _R. J. Mathar_, Jan 22 2013
%t RealDigits[N[Solve[x^3 + x^2 - 1 == 0, x] [[1]] [[1, 2]], 111]] [[1]]
%t RealDigits[x /. FindRoot[x^3 + x^2 == 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* _Harvey P. Dale_, Nov 23 2012 *)
%o (PARI) solve(x=0, 1, x^3+x^2-1)
%o (PARI) polrootsreal(x^3 + x^2 - 1)[1] \\ _Charles R Greathouse IV_, Jul 23 2020
%Y Cf. A060006 (inverse), A210462, A210463.
%K nonn,cons
%O 0,1
%A _Cino Hilliard_, Oct 09 2002
%E More terms from _Robert G. Wilson v_, Oct 10 2002
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