A381124 -id:A381124 - cp's OEIS frontend

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

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

A369346 Continued fraction expansion of the real root of x^3 - x^2 - 1 = 0.

Original entry on oeis.org

1, 2, 6, 1, 3, 5, 4, 22, 1, 1, 4, 1, 2, 84, 1, 3, 1, 6, 1, 3, 1, 9, 1, 1, 1, 1, 19, 3, 1, 2, 1, 5, 1, 5, 2, 2, 1, 1, 1, 1, 76, 6, 8, 1, 1, 5, 1, 5, 1, 1, 25, 1, 2, 1, 116, 2, 1, 8, 1, 1, 3, 1, 53, 5, 276, 2, 1, 1, 1, 3, 3, 2, 1, 1, 4, 13, 1, 1, 1, 4, 1, 1, 1, 9, 9, 1, 1, 9, 6, 1, 2, 32

Offset: 0

Patrick McKinley, Jan 20 2024

Patrick McKinley, Table of n, a(n) for n = 0..12174
Eric Weisstein's World of Mathematics, Supergolden Ratio.

Cf. A092526 (decimal expansion), A381124, A381125 (convergents).

ContinuedFraction[x/.First[Solve[x^3-x^2-1==0,x]],92] (* Stefano Spezia, Jan 21 2024 *)

\p100 \\ realprecision
contfrac(solve(x = 1, 2, x^3 - x^2 - 1),, 80) \\ Hugo Pfoertner, Jan 21 2024

/* The "test" calculation evaluates the cubic to confirm the calculation of the root. */
define iter(frac)
{j = 0
 while(frac > 1){
   frac -= 1;
   j+=1}
 j
 return 1/frac}
scale=12578
f=(1+(e(l(((29+3*sqrt(93))/2))/3))+(e(l(((29-3*sqrt(93))/2))/3)))/3
psi=f
test=(psi-1)*psi*psi-1
for(i=0;i<12175;i++)f=iter(f)

Offset changed by Andrew Howroyd, Feb 14 2025

A381125 Denominators of the convergents to the supergolden ratio.

Original entry on oeis.org

1, 2, 13, 15, 58, 305, 1278, 28421, 29699, 58120, 262179, 320299, 902777, 76153567, 77056344, 307322599, 384378943, 2613596257, 2997975200, 11607521857, 14605497057, 143056995370, 157662492427, 300719487797, 458381980224, 759101468021, 14881309872623

Offset: 0

Eric W. Weisstein, Feb 14 2025

Eric Weisstein's World of Mathematics, Convergent.
Eric Weisstein's World of Mathematics, Denominator.
Eric Weisstein's World of Mathematics, Supergolden Ratio.

Cf. A092526 (decimal digits), A369346 (continued fraction).

Cf. A381124 (numerators of convergents).

Denominator[Convergents[Root[-1 - #^2 + #^3&, 1, 0], 20]]

cp's OEIS Frontend

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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A369346 Continued fraction expansion of the real root of x^3 - x^2 - 1 = 0.

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A381125 Denominators of the convergents to the supergolden ratio.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A369346 Continued fraction expansion of the real root of x^3 - x^2 - 1 = 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

bc

Extensions

A381125 Denominators of the convergents to the supergolden ratio.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

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