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.
Equals -(1/3) + (1/3)*(25/2 - (3*sqrt(69))/2)^(1/3) + (1/3)*((1/2)*(25 + 3*sqrt(69)))^(1/3).
EXAMPLE
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)
(PARI) polrootsreal(x^3 + x^2 - 1)[1] \\ Charles R Greathouse IV, Jul 23 2020
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Cino Hilliard, Oct 09 2002
EXTENSIONS
More terms from Robert G. Wilson v, Oct 10 2002
STATUS
approved