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

1,2

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

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. 238).

Ed Pegg Jr., Images based on the supergolden ratio

Michael Penn, What is the super-golden ratio??, YouTube video, 2022.

Wikipedia, Pisot number

Wikipedia, Supergolden ratio

Index entries for algebraic numbers, degree 3

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

Example

1.46557123187676802665673122521993910802557756847228570164318311124926...

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 *)

Prog

(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

Crossrefs

Cf. A088559, A075778, A076725, A000930, A263719.

Other Pisot numbers: A060006, A086106, A228777, A293506, A293508, A293509, A293557.

Sequence in context: A200497 A271365 A088559 * A307463 A243396 A140243

Adjacent sequences: A092523 A092524 A092525 * A092527 A092528 A092529

Keyword

nonn,cons,easy

Author

N. J. A. Sloane, Apr 07 2004

Status

approved