A293508 -id:A293508 - cp's OEIS frontend

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

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, 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, 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

Offset: 1

Fabian Rothelius, Mar 14 2001

Has been also called the silver number, also the plastic number.
This is the smallest Pisot-Vijayaraghavan number.
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
Sometimes denoted by the symbol rho. - Ed Pegg Jr, Feb 01 2019
Also the solution of 1/x + 1/(1+x+x^2) = 1. - Clark Kimberling, Jan 02 2020
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
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
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

			1.32471795724474602596090885447809734...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
Midhat J. Gazalé, Gnomon: From Pharaohs to Fractals, Princeton University Press, Princeton, NJ, 1999, see Chap. VII.
Donald E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4, p. 236.
Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275 No. 5, November 1996, p. 118.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Alex Bellos, The golden ratio has spawned a beautiful new curve: the Harriss spiral, The Guardian, Jan 13 2015.
Gamaliel Cerda-Morales, New Identities for Padovan Numbers, arXiv:1904.05492 [math.CO], 2019.
Brady Haran and Edmund Harriss, The Plastic Ratio, Numberphile video (2019).
Ed Pegg Jr., Pictures based on the plastic constant
Simon Plouffe, Smallest Pisot-Vijayaraghavan number to 50000 digits
Simon Plouffe, The Smallest Pisot-Vijayaraghavan number
F. Rothelius, Formulae.
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
Dom Hans van der Laan, Le nombre plastique: Quinze leçons sur l’ordonnance architectonique, Brill Academic Pub., Leiden, 1960.
Michel Waldschmidt, Lectures on Multiple Zeta Values, IMSC 2011.
Eric Weisstein's World of Mathematics, Maverick Graph.
Eric Weisstein's World of Mathematics, Pisot Number.
Eric Weisstein's World of Mathematics, Plastic Constant.
Wikipedia, Plastic number.
Index entries for algebraic numbers, degree 3

Cf. A001622. A072117 gives continued fraction.
Cf. A006888, A010527, A051016, A051017, A084252, A075778 (inverse), A126772.
Other Pisot numbers: A086106, A092526, A228777, A293506, A293508, A293509, A293557.
Cf. A002620, A006721, A134816.

SetDefaultRealField(RealField(100)); ((3+Sqrt(23/3))/6)^(1/3) + ((3-Sqrt(23/3))/6)^(1/3); // G. C. Greubel, Mar 15 2019

(1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3) ; evalf(%,130) ; # R. J. Mathar, Jan 22 2013

RealDigits[ Solve[x^3 - x - 1 == 0, x][[1, 1, 2]], 10, 111][[1]] (* Robert G. Wilson v, Sep 30 2009 *)
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 *)
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 *)

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

(1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3) \\ Altug Alkan, Apr 10 2016

polrootsreal(x^3-x-1)[1] \\ Charles R Greathouse IV, Aug 28 2016

default(realprecision, 110); digits(floor(solve(x=1, 2, x^3 - x - 1)*10^105)) /* Michael Somos, Mar 24 2023 */

numerical_approx(((3+sqrt(23/3))/6)^(1/3) + ((3-sqrt(23/3))/6)^(1/3), digits=100) # G. C. Greubel, Mar 15 2019

Equals (1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3). - Henry Bottomley, May 22 2003
Equals CubeRoot(1 + CubeRoot(1 + CubeRoot(1 + CubeRoot(1 + ...)))). - Gerald McGarvey, Nov 26 2004
Equals sqrt(1+1/sqrt(1+1/sqrt(1+1/sqrt(1+...)))). - Gerald McGarvey, Mar 18 2006
Equals (1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3). - Eric Desbiaux, Oct 17 2008
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
Equals sqrt(Phi) = sqrt(1.754877666246...) (see A109134). - Philippe Deléham, Sep 29 2020
Equals cosh(arccosh(3*c)/3)/c, where c = sqrt(3)/2 (A010527). - Amiram Eldar, May 15 2021
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

Edited and extended by Robert G. Wilson v, Aug 03 2002

Removed incorrect comments, Joerg Arndt, Apr 10 2016

A092526 Decimal expansion of (2/3)cos( (1/3)arccos(29/2) ) + 1/3, the real root of x^3 - x^2 - 1.

Original entry on oeis.org

1, 4, 6, 5, 5, 7, 1, 2, 3, 1, 8, 7, 6, 7, 6, 8, 0, 2, 6, 6, 5, 6, 7, 3, 1, 2, 2, 5, 2, 1, 9, 9, 3, 9, 1, 0, 8, 0, 2, 5, 5, 7, 7, 5, 6, 8, 4, 7, 2, 2, 8, 5, 7, 0, 1, 6, 4, 3, 1, 8, 3, 1, 1, 1, 2, 4, 9, 2, 6, 2, 9, 9, 6, 6, 8, 5, 0, 1, 7, 8, 4, 0, 4, 7, 8, 1, 2, 5, 8, 0, 1, 1, 9, 4, 9, 0, 9, 2, 7, 0, 0, 6, 4, 3, 8

Offset: 1

N. J. A. Sloane, Apr 07 2004

This is the limit x of the ratio N(n+1)/N(n) for n -> infinity of the Narayana sequence N(n) = A000930(n). The real root of x^3 - x^2 - 1. See the formula section. - Wolfdieter Lang, Apr 24 2015
This is the fourth smallest Pisot number. - Iain Fox, Oct 13 2017
Sometimes called the supergolden ratio or Narayana's cows constant, and denoted by the symbol psi. - Ed Pegg Jr, Feb 01 2019

			1.46557123187676802665673122521993910802557756847228570164318311124926...

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.3.
Paul J. Nahin, The Logician and the Engineer, How George Boole and Claude Shannon Created the Information Age, Princeton University Press, Princeton and Oxford, 2013, Chap. 7: Some Combinational Logic Examples, Section 7.1: Channel Capacity, Shannon's Theorem, and Error-Detection Theory, page 120.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Simon Baker, Exceptional digit frequencies and expansions in non-integer bases, arXiv:1711.10397 [math.DS], 2017. See the beta(2) constant pp. 3-4.
H. R. P. Ferguson, On a Generalization of the Fibonacci Numbers Useful in Memory Allocation Schema or All About the Zeroes of Z^k - Z^{k - 1} - 1, k > 0, Fibonacci Quarterly, Volume 14, Number 3, October, 1976 (see Table 2, p. 236).
Ed Pegg Jr., Images based on the supergolden ratio
Michael Penn, What is the super-golden ratio??, YouTube video, 2022.
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 59.
Eric Weisstein's World of Mathematics, Pisot Number.
Eric Weisstein's World of Mathematics, Supergolden Ratio.
Wikipedia, Pisot number
Wikipedia, Supergolden ratio
Index entries for algebraic numbers, degree 3

Cf. A088559, A075778, A076725, A000930, A263719.
Other Pisot numbers: A060006, A086106, A228777, A293506, A293508, A293509, A293557.
Cf. A381124 (numerators of convergents).
Cf. A381125 (denominators of convergents).

RealDigits[(2 Cos[ ArcCos[ 29/2]/3] + 1)/3, 10, 111][[1]] (* Robert G. Wilson v, Apr 12 2004 *)
RealDigits[ Solve[ x^3 - x^2 - 1 == 0, x][[1, 1, 2]], 10, 111][[1]] (* Robert G. Wilson v, Oct 10 2013 *)

allocatemem(932245000); default(realprecision, 20080); x=solve(x=1, 2, x^3 - x^2 - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b092526.txt", n, " ", d));  \\ Harry J. Smith, Jun 21 2009

The real root of x^3 - x^2 - 1. - Franklin T. Adams-Watters, Oct 12 2006
The only real irrational root of x^4-x^2-x-1 (-1 is also a root). [Nahim]
Equals (2/3)*cos( (1/3)*arccos(29/2) ) + 1/3.
Equals 1 + A088559.
Equals (1/6)*(116+12*sqrt(93))^(1/3) + 2/(3*(116+12*sqrt(93))^(1/3)) + 1/3. - Vaclav Kotesovec, Dec 18 2014
Equals 1/A263719. - Alois P. Heinz, Apr 15 2018
Equals (1 + 1/r + r)/3 where r = ((29 + sqrt(837))/2)^(1/3). - Peter Luschny, Apr 04 2020
Equals (1/3)*(1 + ((1/2)*(29 + (3*sqrt(93))))^(1/3) + ((1/2)*(29 - 3*sqrt(93)))^(1/3)). See A075778. - Wolfdieter Lang, Aug 17 2022

A293506 Decimal expansion of real root of x^5 - x^4 - x^2 - 1.

Original entry on oeis.org

1, 5, 7, 0, 1, 4, 7, 3, 1, 2, 1, 9, 6, 0, 5, 4, 3, 6, 2, 9, 1, 0, 6, 6, 5, 4, 3, 5, 1, 3, 7, 1, 2, 6, 5, 5, 3, 8, 7, 3, 1, 3, 1, 6, 0, 7, 4, 2, 4, 5, 2, 7, 4, 3, 6, 9, 3, 1, 6, 5, 4, 8, 7, 7, 8, 9, 7, 3, 3, 0, 6, 6, 1, 5, 4, 4, 1, 6, 2, 3, 2, 0, 2, 2, 2, 7, 6

Offset: 1

Iain Fox, Oct 10 2017

This root is also the ninth smallest of the Pisot numbers.

The ratio of successive terms of A122115 converges to this number.

			1.570147312196054362910665...

Iain Fox, Table of n, a(n) for n = 1..20000
Hans Walser, Funktionenfolge [in German]
Eric Weisstein's World of Mathematics, Pisot Number.
Wikipedia, Pisot-Vijayaraghavan number
Index entries for algebraic numbers, degree 5

Cf. A060006, A086106, A228777, A092526, A293508, A293509, A293557, A122115, A293560 (inverse).

First@ RealDigits[Root[#^5 - #^4 - #^2 - 1 &, 1], 10, 87] (* Michael De Vlieger, Oct 23 2017 *)

solve(x=1, 2, x^5 - x^4 - x^2 - 1) \\ Michel Marcus, Oct 11 2017

default(realprecision, 20080); x=solve(x=1, 2, x^5 - x^4 - x^2 - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b293506.txt", n, " ", d));

More terms from Andrey Zabolotskiy, Oct 12 2017

A293509 Decimal expansion of real root of x^5 - x^3 - x^2 - x - 1.

Original entry on oeis.org

1, 5, 3, 4, 1, 5, 7, 7, 4, 4, 9, 1, 4, 2, 6, 6, 9, 1, 5, 4, 3, 5, 9, 7, 0, 0, 7, 6, 1, 0, 9, 3, 7, 5, 7, 0, 1, 8, 8, 2, 5, 4, 5, 0, 3, 8, 5, 1, 6, 5, 9, 5, 1, 3, 5, 3, 6, 8, 5, 3, 1, 8, 6, 3, 0, 0, 8, 0, 6, 3, 0, 2, 3, 2, 1, 4, 0, 8, 2, 2, 8, 1, 4, 3, 6, 7, 8

Offset: 1

Iain Fox, Oct 11 2017

This root is also the sixth smallest of the Pisot numbers.

			1.53415774491426691543597007610937570188254503851659513536853186300806302321...

Iain Fox, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Pisot Number.
Wikipedia, Pisot-Vijayaraghavan number
Index entries for algebraic numbers, degree 5

Cf. A060006, A086106, A228777, A092526, A293506, A293508, A293557.

RealDigits[ Solve[ x^5 - x^3 - x^2 - x - 1 == 0, x, WorkingPrecision -> 111][[-1, 1, 2]], 10, 111][[1]] (* Robert G. Wilson v, Nov 04 2017 *)

solve(x=1, 2, x^5 - x^3 - x^2 - x - 1) \\ Michel Marcus, Oct 13 2017

default(realprecision, 20080); x=solve(x=1, 2, x^5 - x^3 - x^2 - x - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b293509.txt", n, " ", d)); \\ Iain Fox, Oct 23 2017

polrootsreal(x^5 - x^3 - x^2 - x - 1)[1] \\ Charles R Greathouse IV, Nov 04 2017

A293557 Decimal expansion of real root of x^7 - x^6 - x^5 + x^2 - 1.

Original entry on oeis.org

1, 5, 4, 5, 2, 1, 5, 6, 4, 9, 7, 3, 2, 7, 5, 5, 2, 4, 3, 2, 5, 2, 5, 5, 0, 6, 2, 4, 1, 0, 5, 1, 1, 6, 1, 1, 9, 6, 9, 1, 4, 7, 0, 0, 5, 5, 3, 6, 4, 2, 3, 3, 1, 2, 3, 5, 6, 0, 6, 1, 0, 7, 2, 5, 4, 9, 8, 2, 1, 1, 5, 8, 8, 1, 6, 6, 5, 3, 3, 1, 2, 0, 5, 0, 4, 3, 1

Offset: 1

Iain Fox, Oct 11 2017

This root is also the seventh smallest of the Pisot numbers.

			1.545215649732755243252550...

Iain Fox, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Pisot Number.
Wikipedia, Pisot-Vijayaraghavan number
Index entries for algebraic numbers, degree 7.

Cf. A060006, A086106, A228777, A092526, A293508, A293509, A374002, A293506, A374003.

First[RealDigits[Root[#^7 - #^6 - #^5 + #^2 - 1 &, 1], 10, 100]] (* Paolo Xausa, Jun 25 2024 *)

solve(x=1, 2, x^7 - x^6 - x^5 + x^2 - 1) \\ Michel Marcus, Oct 13 2017

{ default(realprecision, 20080); x=solve(x=1, 2, x^7 - x^6 - x^5 + x^2 - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b293557.txt", n, " ", d)); }

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

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A092526 Decimal expansion of (2/3)*cos( (1/3)*arccos(29/2) ) + 1/3, the real root of x^3 - x^2 - 1.

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A293506 Decimal expansion of real root of x^5 - x^4 - x^2 - 1.

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A293509 Decimal expansion of real root of x^5 - x^3 - x^2 - x - 1.

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A293557 Decimal expansion of real root of x^7 - x^6 - x^5 + x^2 - 1.

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A374002 Decimal expansion of the positive real root of x^6 - 2*x^5 + x^4 - x^2 + x - 1.

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A092526 Decimal expansion of (2/3)cos( (1/3)arccos(29/2) ) + 1/3, the real root of x^3 - x^2 - 1.