A298813 - OEIS

%I #40 Jan 04 2024 12:42:45

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

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

%U 1,5,5,0,1,5,1,6,4,0,8,7,8,3,1,1,0,8

%N Decimal expansion of the greatest real zero of x^4 - 2*x^2 - x + 1.

%C Let (d(n)) = (1,0,1,0,1,0,1,...), s(n) = sqrt(s(n-1) + d(n)) for n > 0, and s(0) = 1.

%C Then s(2n) -> 1.49021612009995..., as in A298813;

%C and s(2n+1) -> 1.22074408..., as in A060007.

%C Let (e(n)) = (0,1,0,1,0,1,0,...), t(n) = sqrt(t(n-1) + e(n)) for n > 0, and t(0) = 1.

%C Then t(2n) -> 1.22074408..., as in A060007;

%C and t(2n+1) -> 1.49021612009995..., as in A298813.

%C The four solutions are: x1, this one; x2, the least A072223; and the two complex ones x3=-1.007552359378... + 0.513115795597...*i and x4, its complex conjugate; Re(x3) = Re(x4) = -(x1+x2)/2; Im(x3) = -Im(x4) = sqrt(1/(x1*x2) - Re(x3)^2). - _Andrea Pinos_, Sep 20 2023

%H Clark Kimberling, <a href="/A298813/b298813.txt">Table of n, a(n) for n = 1..10000</a>

%F Equals sqrt((1 + 2*cos(arccos(155/128)/3))/3) + sqrt(2/3 - 2*cos(arccos(155/128)/3)/3 + sqrt(3/(1 + 2*cos(arccos(155/128)/3)))/4). - _Vaclav Kotesovec_, Sep 21 2023

%F Equals sqrt(1/3 + s/9 + 1/s) + sqrt(2/3 - s/9 - 1/s + 1 / (4 * sqrt(1/3 + s/9 + 1/s))) where s = (4185/128 + sqrt(5570289/16384))^(1/3). - _Michal Paulovic_, Dec 30 2023

%e 1.49021612009995...

%t r = x /. NSolve[x^4 - 2 x^2 - x + 1 == 0, x, 100][[4]];

%t RealDigits[r][[1]]; (* A298813 *)

%t RealDigits[Root[x^4-2x^2-x+1,2],10,120][[1]] (* _Harvey P. Dale_, May 02 2022 *)

%o (PARI) solve(x=1, 2, x^4 - 2*x^2 - x + 1) \\ _Michel Marcus_, Nov 05 2018

%Y Cf. A060007, A072223, A298814, A298815.

%K nonn,easy,cons

%O 1,2

%A _Clark Kimberling_, Feb 13 2018