A298813 Decimal expansion of the greatest real zero of x^4 - 2*x^2 - x + 1.

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

Offset: 1

Author

Clark Kimberling, Feb 13 2018

Keywords

• nonn
• easy
• cons

Comments

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.

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

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

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.

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

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

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

Examples

			1.49021612009995...
		

Links

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

Crossrefs

Cf. A060007, A072223, A298814, A298815.

Programs

Mathematica

r = x /. NSolve[x^4 - 2 x^2 - x + 1 == 0, x, 100][[4]];
RealDigits[r][[1]]; (* A298813 *)
RealDigits[Root[x^4-2x^2-x+1,2],10,120][[1]] (* Harvey P. Dale, May 02 2022 *)

PARI

solve(x=1, 2, x^4 - 2*x^2 - x + 1) \\ Michel Marcus, Nov 05 2018

Formula

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

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