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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.

%I A092526 #123 Jun 15 2025 23:16:37
%S A092526 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,
%T A092526 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,
%U A092526 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
%N A092526 Decimal expansion of (2/3)*cos( (1/3)*arccos(29/2) ) + 1/3, the real root of x^3 - x^2 - 1.
%C A092526 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
%C A092526 This is the fourth smallest Pisot number. - _Iain Fox_, Oct 13 2017
%C A092526 Sometimes called the supergolden ratio or Narayana's cows constant, and denoted by the symbol psi. - _Ed Pegg Jr_, Feb 01 2019
%D A092526 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.3.
%D A092526 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.
%H A092526 Harry J. Smith, <a href="/A092526/b092526.txt">Table of n, a(n) for n = 1..20000</a>
%H A092526 Simon Baker, <a href="https://arxiv.org/abs/1711.10397">Exceptional digit frequencies and expansions in non-integer bases</a>, arXiv:1711.10397 [math.DS], 2017. See the beta(2) constant pp. 3-4.
%H A092526 H. R. P. Ferguson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/14-3/ferguson.pdf">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</a>, Fibonacci Quarterly, Volume 14, Number 3, October, 1976 (see Table 2, p. 236).
%H A092526 Ed Pegg Jr., <a href="/A092526/a092526.jpg">Images based on the supergolden ratio</a>
%H A092526 Michael Penn, <a href="https://www.youtube.com/watch?v=X9DpdomPRvg">What is the super-golden ratio??</a>, YouTube video, 2022.
%H A092526 Mario Raso, <a href="https://iris.uniroma1.it/handle/11573/1732819">Integer Sequences in Cryptography: A New Generalized Family and its Application</a>, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 59.
%H A092526 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PisotNumber.html">Pisot Number</a>.
%H A092526 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SupergoldenRatio.html">Supergolden Ratio</a>.
%H A092526 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pisot-Vijayaraghavan_number#Small_Pisot_numbers">Pisot number</a>
%H A092526 Wikipedia, <a href="https://en.wikipedia.org/wiki/Supergolden_ratio">Supergolden ratio</a>
%H A092526 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%F A092526 The real root of x^3 - x^2 - 1. - _Franklin T. Adams-Watters_, Oct 12 2006
%F A092526 The only real irrational root of x^4-x^2-x-1 (-1 is also a root). [Nahim]
%F A092526 Equals (2/3)*cos( (1/3)*arccos(29/2) ) + 1/3.
%F A092526 Equals 1 + A088559.
%F A092526 Equals (1/6)*(116+12*sqrt(93))^(1/3) + 2/(3*(116+12*sqrt(93))^(1/3)) + 1/3. - _Vaclav Kotesovec_, Dec 18 2014
%F A092526 Equals 1/A263719. - _Alois P. Heinz_, Apr 15 2018
%F A092526 Equals (1 + 1/r + r)/3 where r = ((29 + sqrt(837))/2)^(1/3). - _Peter Luschny_, Apr 04 2020
%F A092526 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
%e A092526 1.46557123187676802665673122521993910802557756847228570164318311124926...
%t A092526 RealDigits[(2 Cos[ ArcCos[ 29/2]/3] + 1)/3, 10, 111][[1]] (* _Robert G. Wilson v_, Apr 12 2004 *)
%t A092526 RealDigits[ Solve[ x^3 - x^2 - 1 == 0, x][[1, 1, 2]], 10, 111][[1]] (* _Robert G. Wilson v_, Oct 10 2013 *)
%o A092526 (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
%Y A092526 Cf. A088559, A075778, A076725, A000930, A263719.
%Y A092526 Other Pisot numbers: A060006, A086106, A228777, A293506, A293508, A293509, A293557.
%Y A092526 Cf. A381124 (numerators of convergents).
%Y A092526 Cf. A381125 (denominators of convergents).
%K A092526 nonn,cons,easy
%O A092526 1,2
%A A092526 _N. J. A. Sloane_, Apr 07 2004