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

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

Author

N. J. A. Sloane, Apr 07 2004

Keywords

• nonn
• cons
• easy

Comments

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

Examples

			1.46557123187676802665673122521993910802557756847228570164318311124926...
		

References

• 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.

Links

• 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

Crossrefs

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).

Programs

Mathematica

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 *)

PARI

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

Formula

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