%I #22 Oct 27 2023 10:30:52
%S 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,
%T 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,
%U 7,0,0,6,8,0,7,8,9,0,4,9,9,7,5,3,8,2
%N Decimal expansion of the greatest real zero of x^4 - 2*x^2 - x - 1.
%C 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.
%C Then s(2n) -> 1.9263032199..., as in A298852;
%C s(2n+1) -> 1.710644095..., as in A298853.
%H Clark Kimberling, <a href="/A298853/b298853.txt">Table of n, a(n) for n = 1..10000</a>
%H Simon Baker, <a href="https://doi.org/10.1016/j.jnt.2014.08.003">On small bases which admit countably many expansions</a>, Journal of Number Theory, Volume 147, February 2015, Pages 515-532.
%H Nikita Sidorov, <a href="https://doi.org/10.1016/j.jnt.2008.11.003">Expansions in non-integer bases: Lower, middle and top orders</a>, Journal of Number Theory, Volume 129, Issue 4, April 2009, Pages 741-754. See Proposition 2.4 p. 744.
%H Yuru Zou, Derong Kong, <a href="https://doi.org/10.1016/j.jnt.2015.06.017">On a problem of countable expansions</a>, Journal of Number Theory, Volume 158, January 2016, Pages 134-150. See Theorem 1.1 p. 135.
%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%e Greatest real zero = 1.710644095...
%t r = x /. NSolve[x^4 - 2 x^2 - x - 1 == 0, x, 10000][[4]];
%t RealDigits[r][[1]]; (* A298853 *)
%t RealDigits[Root[ x^4-2*x^2-x-1,2],10,120][[1]] (* _Harvey P. Dale_, May 23 2019 *)
%o (PARI) solve(x=1, 2, x^4-2*x^2-x-1) \\ _Michel Marcus_, Apr 14 2020
%o (PARI) polrootsreal(x^4 - 2*x^2 - x - 1)[2] \\ _Charles R Greathouse IV_, May 15 2020
%Y Cf. A298852.
%K cons,nonn,easy
%O 1,2
%A _Clark Kimberling_, Feb 13 2018