cp's OEIS Frontend

A060007 Decimal expansion of the positive real root of x^4 - x - 1.

%I A060007 #73 Jul 17 2025 00:46:31
%S A060007 1,2,2,0,7,4,4,0,8,4,6,0,5,7,5,9,4,7,5,3,6,1,6,8,5,3,4,9,1,0,8,8,3,1,
%T A060007 9,1,4,4,3,2,4,8,9,0,8,6,2,4,8,6,3,5,2,1,4,2,8,8,2,4,4,4,5,3,0,4,9,7,
%U A060007 1,0,0,0,8,5,2,2,5,9,1,3,5,0,2,5,3,0,9,5,5,2,1,8,6,9,9,6,2,8,6,2,5,7,4,0,1
%N A060007 Decimal expansion of the positive real root of x^4 - x - 1.
%C A060007 Original name: Decimal expansion of v_4, where v_n is the smallest, positive, real solution to the equation (v_n)^n = v_n + 1.
%C A060007 v_2 = A001622 - 1. [Corrected by _M. F. Hasler_, Jul 12 2025]
%C A060007 v_3 = A060006, a.k.a. plastic constant, real root of x^3 - x - 1. - _M. F. Hasler_, Jul 12 2025
%C A060007 A Perron number of the 4th degree polynomial (see Boys and Wu). - _R. J. Mathar_, Mar 19 2011
%C A060007 This number is not ruler-and-compass constructible because x^4-x-1 and its resolvent x^3+4x+1 are irreducible over the rationals. - _Jean-François Alcover_, Aug 31 2015
%C A060007 The other (negative) real root -0.724491959... is -A356032. The first of the pair of complex conjugate roots is obtained by negating in the formula for v_4 below sqrt(2*u) and sqrt(u), giving -0.2481260628... - 1.0339820609...*i. - _Wolfdieter Lang_, Aug 27 2022
%C A060007 The sequence a(n) = v_4^((n^2-n)/2) satisfies the Somos-4 recursion a(n+2)*a(n-2) = a(n+1)*a(n-1) + a(n)^2 for all n in Z. - _Michael Somos_, Mar 24 2023
%H A060007 Vincenzo Librandi, <a href="/A060007/b060007.txt">Table of n, a(n) for n = 1..1000</a>
%H A060007 David W. Boys, <a href="http://dx.doi.org/10.1090/S0025-5718-1985-0790657-8">The maximal modulus of an algebraic integer</a>, Math. Comp. 45 (1985) 243-249, table page S18.
%H A060007 F. Rothelius, <a href="https://web.archive.org/web/20010628042609/http://w1.875.telia.com/~u87509703/mathez/v2v3v4.gif">Formulae</a>
%H A060007 Qiang Wu, <a href="http://dx.doi.org/10.1090/S0025-5718-10-02345-8">The smallest Perron numbers</a>, Math. Comp. 79 (2010) 2387-2394
%H A060007 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%F A060007 Equals (1 + (1 + (1 + (1 + (1 + ...)^(1/4))^(1/4))^(1/4))^(1/4))^(1/4). - _Ilya Gutkovskiy_, Dec 15 2017
%F A060007 v_4 = (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)), ep = (-1 + sqrt(3)*i)/2 and i = sqrt(-1).
%F A060007 For the trigonometric equivalent u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/16)* sqrt(3))). - _Wolfdieter Lang_, Aug 27 2022
%F A060007 Equals 1 + Sum_{n >= 1} (1/4)^n*(Product_{j=1..n-1} 1 + n - 4*j)/n!. - _Antonio Graciá Llorente_, Dec 13 2024
%F A060007 Equals exp(A202540) = sqrt(A298813). - _Hugo Pfoertner_, Dec 14 2024
%e A060007 v_4 = 1.220744084605759475361685349...
%p A060007 r:=(108+12*sqrt(849))^(1/3): (sqrt(12/sqrt(-8/r+r/6)+48/r-r) + sqrt(-48/r+r))/(2*sqrt(6)): evalf(%,105); # _Vaclav Kotesovec_, Oct 12 2013
%t A060007 RealDigits[x/.FindRoot[x^4==x+1,{x,1},WorkingPrecision->120]][[1]] (* _Harvey P. Dale_, Jul 11 2012 *)
%t A060007 Root[ #^4 - # - 1&, 2] // RealDigits[#, 10, 105]& // First (* _Jean-François Alcover_, Mar 04 2013 *)
%o A060007 (PARI) default(realprecision, 110); digits(floor(solve(x=1, 2, x^4 - x - 1)*10^105)) /* _Michael Somos_, Mar 22 2023 */
%Y A060007 Cf. A001622 (golden ratio, root of x^2 - x - 1), A060006 (plastic number, root of x^3 - x - 1), A202540 (log thereof), A160155 (root of x^5 - x - 1), A356032 (root of x^4 + x - 1), A006720, A298813.
%K A060007 cons,nice,nonn
%O A060007 1,2
%A A060007 _Fabian Rothelius_, Mar 14 2001
%E A060007 More terms from _Benoit Cloitre_, Jan 11 2003
%E A060007 Simplified definition from _M. F. Hasler_, Jul 12 2025