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A060006 Decimal expansion of real root of x^3 - x - 1 (the plastic constant).

%I A060006 #216 Mar 16 2025 10:13:11
%S A060006 1,3,2,4,7,1,7,9,5,7,2,4,4,7,4,6,0,2,5,9,6,0,9,0,8,8,5,4,4,7,8,0,9,7,
%T A060006 3,4,0,7,3,4,4,0,4,0,5,6,9,0,1,7,3,3,3,6,4,5,3,4,0,1,5,0,5,0,3,0,2,8,
%U A060006 2,7,8,5,1,2,4,5,5,4,7,5,9,4,0,5,4,6,9,9,3,4,7,9,8,1,7,8,7,2,8,0,3,2,9,9,1
%N A060006 Decimal expansion of real root of x^3 - x - 1 (the plastic constant).
%C A060006 Has been also called the silver number, also the plastic number.
%C A060006 This is the smallest Pisot-Vijayaraghavan number.
%C A060006 The name "plastic number" goes back to the Dutch Benedictine monk and architect Dom Hans van der Laan, who gave this name 4 years after the discovery of the number by the French engineer Gérard Cordonnier in 1924, who used the name "radiant number". - _Hugo Pfoertner_, Oct 07 2018
%C A060006 Sometimes denoted by the symbol rho. - _Ed Pegg Jr_, Feb 01 2019
%C A060006 Also the solution of 1/x + 1/(1+x+x^2) = 1. - _Clark Kimberling_, Jan 02 2020
%C A060006 Given any complex p such that real(p)>-1, this constant is the only real solution of the equation z^p+z^(p+1)=z^(p+3), and the only attractor of the complex mapping z->M(z,p), where M(z,p)=(z^p+z^(p+1))^(1/(p+3)), convergent from any complex plane point. - _Stanislav Sykora_, Oct 14 2021
%C A060006 The Pisot-Vijayaraghavan numbers were named after the French mathematician Charles Pisot (1910-1984) and the Indian mathematician Tirukkannapuram Vijayaraghavan (1902-1955). - _Amiram Eldar_, Apr 02 2022
%C A060006 The sequence a(n) = v_3^floor(n^2/4) where v_n is the smallest, positive, real solution to the equation (v_n)^n = v_n + 1 satisfies the Somos-5 recursion a(n+3)*a(n-2) = a(n+2)*a(n-1) + a(n+1)*a(n) for all n in Z. Also true if floor is removed. - _Michael Somos_, Mar 24 2023
%D A060006 Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
%D A060006 Midhat J. Gazalé, Gnomon: From Pharaohs to Fractals, Princeton University Press, Princeton, NJ, 1999, see Chap. VII.
%D A060006 Donald E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4, p. 236.
%D A060006 Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275 No. 5, November 1996, p. 118.
%H A060006 Harry J. Smith, <a href="/A060006/b060006.txt">Table of n, a(n) for n = 1..20000</a>
%H A060006 Alex Bellos, <a href="http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/jan/13/golden-ratio-beautiful-new-curve-harriss-spiral">The golden ratio has spawned a beautiful new curve: the Harriss spiral</a>, The Guardian, Jan 13 2015.
%H A060006 Gamaliel Cerda-Morales, <a href="https://arxiv.org/abs/1904.05492">New Identities for Padovan Numbers</a>, arXiv:1904.05492 [math.CO], 2019.
%H A060006 Brady Haran and Edmund Harriss, <a href="https://www.youtube.com/watch?v=PsGUEj4w9Cc">The Plastic Ratio</a>, Numberphile video (2019).
%H A060006 Ed Pegg Jr., <a href="/A060006/a060006.jpg">Pictures based on the plastic constant</a>
%H A060006 Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/pisotv.txt">Smallest Pisot-Vijayaraghavan number to 50000 digits</a>
%H A060006 Simon Plouffe, <a href="https://web.archive.org/web/20150912063058/http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap76.html">The Smallest Pisot-Vijayaraghavan number</a>
%H A060006 F. Rothelius, <a href="https://web.archive.org/web/20010628042609/http://w1.875.telia.com/~u87509703/mathez/v2v3v4.gif">Formulae</a>.
%H A060006 Ian Stewart, <a href="http://wayback.archive.org/web/20120320051231/http://members.fortunecity.com/templarser/padovan.html">Tales of a Neglected Number</a>, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
%H A060006 Ian Stewart, <a href="https://www.jstor.org/stable/24989576">Tales of a Neglected Number</a>, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
%H A060006 Dom Hans van der Laan, <a href="https://brill.com/display/title/133">Le nombre plastique: Quinze leçons sur l’ordonnance architectonique</a>, Brill Academic Pub., Leiden, 1960.
%H A060006 Michel Waldschmidt, <a href="http://www.math.jussieu.fr/~miw/articles/pdf/MZV2011IMSc.pdf">Lectures on Multiple Zeta Values</a>, IMSC 2011.
%H A060006 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaverickGraph.html">Maverick Graph</a>.
%H A060006 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PisotNumber.html">Pisot Number</a>.
%H A060006 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlasticConstant.html">Plastic Constant</a>.
%H A060006 Wikipedia, <a href="https://en.wikipedia.org/wiki/Plastic_number">Plastic number</a>.
%H A060006 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%F A060006 Equals (1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3). - _Henry Bottomley_, May 22 2003
%F A060006 Equals CubeRoot(1 + CubeRoot(1 + CubeRoot(1 + CubeRoot(1 + ...)))). - _Gerald McGarvey_, Nov 26 2004
%F A060006 Equals sqrt(1+1/sqrt(1+1/sqrt(1+1/sqrt(1+...)))). - _Gerald McGarvey_, Mar 18 2006
%F A060006 Equals (1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3). - _Eric Desbiaux_, Oct 17 2008
%F A060006 Equals Sum_{k >= 0} 27^(-k)/k!*(Gamma(2*k+1/3)/(9*Gamma(k+4/3)) - Gamma(2*k-1/3)/(3*Gamma(k+2/3))). - _Robert Israel_, Jan 13 2015
%F A060006 Equals sqrt(Phi) = sqrt(1.754877666246...) (see A109134). - _Philippe Deléham_, Sep 29 2020
%F A060006 Equals cosh(arccosh(3*c)/3)/c, where c = sqrt(3)/2 (A010527). - _Amiram Eldar_, May 15 2021
%F A060006 Equals 1/hypergeom([1/5, 2/5, 3/5, 4/5], [2/4, 3/4, 5/4], -5^5/4^4). - _Gerry Martens_, Mar 16 2025
%e A060006 1.32471795724474602596090885447809734...
%p A060006 (1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3) ; evalf(%,130) ; # _R. J. Mathar_, Jan 22 2013
%t A060006 RealDigits[ Solve[x^3 - x - 1 == 0, x][[1, 1, 2]], 10, 111][[1]] (* _Robert G. Wilson v_, Sep 30 2009 *)
%t A060006 s = Sqrt[23/108]; RealDigits[(1/2 + s)^(1/3) + (1/2 - s)^(1/3), 10, 111][[1]] (* _Robert G. Wilson v_, Dec 12 2017 *)
%t A060006 RealDigits[Root[x^3-x-1,1],10,120][[1]] (* or *) RealDigits[(Surd[9-Sqrt[69],3]+Surd[9+Sqrt[69],3])/(Surd[2,3]Surd[9,3]),10,120][[1]] (* _Harvey P. Dale_, Sep 04 2018 *)
%o A060006 (PARI) allocatemem(932245000); default(realprecision, 20080); x=solve(x=1, 2, x^3 - x - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b060006.txt", n, " ", d)); \\ _Harry J. Smith_, Jul 01 2009
%o A060006 (PARI) (1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3) \\ _Altug Alkan_, Apr 10 2016
%o A060006 (PARI) polrootsreal(x^3-x-1)[1] \\ _Charles R Greathouse IV_, Aug 28 2016
%o A060006 (PARI) default(realprecision, 110); digits(floor(solve(x=1, 2, x^3 - x - 1)*10^105)) /* _Michael Somos_, Mar 24 2023 */
%o A060006 (Magma) SetDefaultRealField(RealField(100)); ((3+Sqrt(23/3))/6)^(1/3) + ((3-Sqrt(23/3))/6)^(1/3); // _G. C. Greubel_, Mar 15 2019
%o A060006 (Sage) numerical_approx(((3+sqrt(23/3))/6)^(1/3) + ((3-sqrt(23/3))/6)^(1/3), digits=100) # _G. C. Greubel_, Mar 15 2019
%Y A060006 Cf. A001622. A072117 gives continued fraction.
%Y A060006 Cf. A006888, A010527, A051016, A051017, A084252, A075778 (inverse), A126772.
%Y A060006 Other Pisot numbers: A086106, A092526, A228777, A293506, A293508, A293509, A293557.
%Y A060006 Cf. A002620, A006721, A134816.
%K A060006 cons,nice,nonn
%O A060006 1,2
%A A060006 _Fabian Rothelius_, Mar 14 2001
%E A060006 Edited and extended by _Robert G. Wilson v_, Aug 03 2002
%E A060006 Removed incorrect comments, _Joerg Arndt_, Apr 10 2016