cp's OEIS Frontend

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

%I A298813 #40 Jan 04 2024 12:42:45
%S A298813 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 A298813 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 A298813 1,5,5,0,1,5,1,6,4,0,8,7,8,3,1,1,0,8
%N A298813 Decimal expansion of the greatest real zero of x^4 - 2*x^2 - x + 1.
%C A298813 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 A298813 Then s(2n) -> 1.49021612009995..., as in A298813;
%C A298813 and s(2n+1) -> 1.22074408..., as in A060007.
%C A298813 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 A298813 Then t(2n) -> 1.22074408..., as in A060007;
%C A298813 and t(2n+1) -> 1.49021612009995..., as in A298813.
%C 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
%H A298813 Clark Kimberling, <a href="/A298813/b298813.txt">Table of n, a(n) for n = 1..10000</a>
%F A298813 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 A298813 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 A298813 1.49021612009995...
%t A298813 r = x /. NSolve[x^4 - 2 x^2 - x + 1 == 0, x, 100][[4]];
%t A298813 RealDigits[r][[1]]; (* A298813 *)
%t A298813 RealDigits[Root[x^4-2x^2-x+1,2],10,120][[1]] (* _Harvey P. Dale_, May 02 2022 *)
%o A298813 (PARI) solve(x=1, 2, x^4 - 2*x^2 - x + 1) \\ _Michel Marcus_, Nov 05 2018
%Y A298813 Cf. A060007, A072223, A298814, A298815.
%K A298813 nonn,easy,cons
%O A298813 1,2
%A A298813 _Clark Kimberling_, Feb 13 2018