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 A075778 Decimal expansion of the real root of x^3 + x^2 - 1. 21
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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 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 REFERENCES 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. LINKS H. R. P. Ferguson, 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, Fibonacci Quarterly, Volume 14, Number 3, October, 1976 (see Table 2 p. 238). FORMULA 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. EXAMPLE x = -(1/3) + (1/3)*(25/2 - (3*sqrt(69))/2)^(1/3) + (1/3)*((1/2)*(25 + 3*sqrt(69)))^(1/3). Equals 0.7548776662466927600495088963585286918946066... MAPLE A075778 := proc()         1/3-root[3](25/2-3*sqrt(69)/2)/3 -root[3](25/2+3*sqrt(69)/2)/3;           -% ; end proc: # R. J. Mathar, Jan 22 2013 MATHEMATICA RealDigits[N[Solve[x^3 + x^2 - 1 == 0, x] [[1]] [[1, 2]], 111]] [[1]] RealDigits[x /. FindRoot[x^3 + x^2 == 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* Harvey P. Dale, Nov 23 2012 *) PROG (PARI) solve(x=0, 1, x^3+x^2-1) CROSSREFS Cf. A060006 (inverse), A210462, A210463. Sequence in context: A289005 A225408 A109134 * A289033 A010510 A258042 Adjacent sequences:  A075775 A075776 A075777 * A075779 A075780 A075781 KEYWORD nonn,cons AUTHOR Cino Hilliard, Oct 09 2002 EXTENSIONS More terms from Robert G. Wilson v, Oct 10 2002 STATUS approved

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Last modified June 7 05:20 EDT 2020. Contains 334837 sequences. (Running on oeis4.)