This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A356032 #33 Jul 14 2025 10:11:09
%S A356032 7,2,4,4,9,1,9,5,9,0,0,0,5,1,5,6,1,1,5,8,8,3,7,2,2,8,2,1,8,7,0,3,6,5,
%T A356032 6,5,7,8,6,4,9,4,4,8,1,3,5,0,0,1,1,0,1,7,2,7,0,3,9,8,0,2,8,4,3,7,4,5,
%U A356032 2,9,0,6,4,7,5,1
%N A356032 Decimal expansion of the positive real root of x^4 + x - 1.
%C A356032 The other real (negative) root is -A060007.
%C A356032 One of the pair of complex conjugate roots is obtained by negating sqrt(2*u) and sqrt(u) in the formula for r below, giving 0.248126062... - 1.033982060...*i.
%C A356032 Also, the absolute value of the negative real root of x^4 - x - 1, cf. A060007. - _M. F. Hasler_, Jul 12 2025
%F A356032 r = (-sqrt(2)*u + sqrt(sqrt(2*u) - 2*u^2))/(2*sqrt(u)), with u = (Ap^(1/3) + ep*Am^(1/3))/3, where Ap = (3/16)*(9 + sqrt(3*283)), Am = (3/16)*(9 - sqrt(3*283)) and ep = (-1 + sqrt(3)*i)/2, with i = sqrt(-1). For the trigonometric version set u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/16)* sqrt(3))).
%F A356032 Equals sqrt(A072223) = 1/A086106 = 1/exp(A202538). - _Hugo Pfoertner_, Jul 13 2025
%e A356032 r = 0.724491959000515611588372282187036565786494481350011017270...
%t A356032 First[RealDigits[x/.N[{x->Root[-1+#1+#1^4 &,2,0]},75]]] (* _Stefano Spezia_, Aug 27 2022 *)
%o A356032 (PARI) solve(x=0, 1, x^4 + x - 1) \\ _Michel Marcus_, Aug 28 2022
%o A356032 (PARI) polrootsreal(x^4 + x - 1)[2] \\ _M. F. Hasler_, Jul 12 2025
%Y A356032 Cf. A060007 (positive root of x^4 - x - 1), A072223, A086106, A202538, A376658.
%K A356032 nonn,cons,easy
%O A356032 0,1
%A A356032 _Wolfdieter Lang_, Aug 27 2022