

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
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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

Table of n, a(n) for n=0..104.
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()
1/3root[3](25/23*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^21)


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



