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

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, 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, 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

Offset: 1

Author

Fabian Rothelius, Mar 14 2001

Keywords

• cons
• nice
• nonn

Comments

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.

v_2 = A001622 - 1. [Corrected by M. F. Hasler, Jul 12 2025]

v_3 = A060006, a.k.a. plastic constant, real root of x^3 - x - 1. - M. F. Hasler, Jul 12 2025

A Perron number of the 4th degree polynomial (see Boys and Wu). - R. J. Mathar, Mar 19 2011

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

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

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

Examples

			v_4 = 1.220744084605759475361685349...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• David W. Boys, The maximal modulus of an algebraic integer, Math. Comp. 45 (1985) 243-249, table page S18.
• F. Rothelius, Formulae
• Qiang Wu, The smallest Perron numbers, Math. Comp. 79 (2010) 2387-2394
• Index entries for algebraic numbers, degree 4

Crossrefs

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.

Programs

Maple

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

Mathematica

RealDigits[x/.FindRoot[x^4==x+1,{x,1},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jul 11 2012 *)
Root[ #^4 - # - 1&, 2] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 04 2013 *)

PARI

default(realprecision, 110); digits(floor(solve(x=1, 2, x^4 - x - 1)*10^105)) /* Michael Somos, Mar 22 2023 */

Formula

Equals (1 + (1 + (1 + (1 + (1 + ...)^(1/4))^(1/4))^(1/4))^(1/4))^(1/4). - Ilya Gutkovskiy, Dec 15 2017

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).

For the trigonometric equivalent u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/16)* sqrt(3))). - Wolfdieter Lang, Aug 27 2022

Equals 1 + Sum_{n >= 1} (1/4)^n*(Product_{j=1..n-1} 1 + n - 4*j)/n!. - Antonio Graciá Llorente, Dec 13 2024

Equals exp(A202540) = sqrt(A298813). - Hugo Pfoertner, Dec 14 2024

Extensions

More terms from Benoit Cloitre, Jan 11 2003

Simplified definition from M. F. Hasler, Jul 12 2025