A298853 Decimal expansion of the greatest real zero of x^4 - 2*x^2 - x - 1.

1, 7, 1, 0, 6, 4, 4, 0, 9, 5, 0, 4, 5, 0, 3, 2, 9, 3, 5, 9, 9, 0, 6, 3, 4, 1, 6, 3, 3, 3, 5, 8, 5, 9, 4, 5, 6, 3, 3, 1, 5, 6, 0, 9, 8, 5, 5, 9, 2, 4, 8, 5, 4, 4, 7, 8, 6, 1, 1, 6, 8, 7, 5, 8, 2, 3, 6, 1, 7, 0, 0, 6, 8, 0, 7, 8, 9, 0, 4, 9, 9, 7, 5, 3, 8, 2

Offset

1,2

Comments

Let (d(n)) = (2,1,2,1,2,1,...), s(n) = (s(n-1) + d(n))^(1/2) for n > 0, and s(0) = 1.

Then s(2n) -> 1.9263032199..., as in A298852;

s(2n+1) -> 1.710644095..., as in A298853.

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

Simon Baker, On small bases which admit countably many expansions, Journal of Number Theory, Volume 147, February 2015, Pages 515-532.

Nikita Sidorov, Expansions in non-integer bases: Lower, middle and top orders, Journal of Number Theory, Volume 129, Issue 4, April 2009, Pages 741-754. See Proposition 2.4 p. 744.

Yuru Zou, Derong Kong, On a problem of countable expansions, Journal of Number Theory, Volume 158, January 2016, Pages 134-150. See Theorem 1.1 p. 135.

Index entries for algebraic numbers, degree 4

Example

Greatest real zero = 1.710644095...

Mathematica

r = x /. NSolve[x^4 - 2 x^2 - x  - 1 == 0, x, 10000][[4]];
RealDigits[r][[1]]; (* A298853 *)
RealDigits[Root[ x^4-2*x^2-x-1, 2], 10, 120][[1]] (* Harvey P. Dale, May 23 2019 *)

Prog

(PARI) solve(x=1, 2, x^4-2*x^2-x-1) \\ Michel Marcus, Apr 14 2020
(PARI) polrootsreal(x^4 - 2*x^2 - x - 1)[2] \\ Charles R Greathouse IV, May 15 2020

Crossrefs

Cf. A298852.

Sequence in context: A179376 A152447 A198611 * A351213 A198212 A217245

Adjacent sequences: A298850 A298851 A298852 * A298854 A298855 A298856

Keyword

cons,nonn,easy

Author

Clark Kimberling, Feb 13 2018

Status

approved