cp's OEIS Frontend

A160389 Decimal expansion of 2*cos(Pi/7).

%I A160389 #116 Aug 31 2025 09:09:33
%S A160389 1,8,0,1,9,3,7,7,3,5,8,0,4,8,3,8,2,5,2,4,7,2,2,0,4,6,3,9,0,1,4,8,9,0,
%T A160389 1,0,2,3,3,1,8,3,8,3,2,4,2,6,3,7,1,4,3,0,0,1,0,7,1,2,4,8,4,6,3,9,8,8,
%U A160389 6,4,8,4,0,8,5,5,8,7,9,9,3,1,0,0,2,7,2,2,9,0,9,4,3,7,0,2,4,8,3,0,6,3,6,6,2
%N A160389 Decimal expansion of 2*cos(Pi/7).
%C A160389 Arises in the approximation of 14-fold quasipatterns by 14 Fourier modes.
%C A160389 Let DTS(n^c) denote the set of languages accepted by a deterministic Turing machine with space n^(o(1)) and time n^(c+o(1)), and let SAT denote the Boolean satisfiability problem. Then (1) SAT is not in DTS(n^c) for any c < 2*cos(Pi/7), and (2) the Williams inference rules cannot prove that SAT is not in DTS(n^c) for any c >= 2*cos(Pi/7). These results also apply to the Boolean satisfiability problem mod m where m is in A085971 except possibly for one prime. - _Charles R Greathouse IV_, Jul 19 2012
%C A160389 rho(7):= 2*cos(Pi/7) is the length ratio (smallest diagonal)/side in the regular 7-gon (heptagon). The algebraic number field Q(rho(7)) of degree 3 is fundamental for the 7-gon. See A187360 for the minimal polynomial C(7, x) of rho(7). The other (larger) diagonal/side ratio in the heptagon is sigma(7) = -1 + rho(7)^2, approx. 2.2469796. (see the decimal expansion in A231187). sigma(7) is the limit of a(n+1)/a(n) for n->infinity for the sequences like A006054 and A077998 which can be considered as analogs of the Fibonacci sequence in the pentagon. Thus sigma(7) plays in the heptagon the role of the golden section in the pentagon. See the P. Steinbach reference. - _Wolfdieter Lang_, Nov 21 2013
%C A160389 An algebraic integer of degree 3 with minimal polynomial x^3 - x^2 - 2x + 1. - _Charles R Greathouse IV_, Nov 12 2014
%C A160389 The other two solutions of the minimal polynomial of rho(7) = 2*cos(Pi/7) are 2*cos(3*Pi/7) and 2*cos(5*Pi/7). See eq. (20) of the W. Lang link. - _Wolfdieter Lang_, Feb 11 2015
%C A160389 The constant is the square root of 3.24697... (cf. A116425). It is the fifth-longest diagonal in the regular 14-gon with unit radius, which equals 2*sin(5*Pi/14). - _Gary W. Adamson_, Feb 14 2022
%D A160389 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.
%H A160389 Harry J. Smith, <a href="/A160389/b160389.txt">Table of n, a(n) for n = 1..20000</a>
%H A160389 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 Theorem 1.1 p. 3.
%H A160389 Sam Buss and Ryan Williams, <a href="http://eccc.hpi-web.de/report/2011/031/">Limits on alternation-trading proofs for time-space lower bounds</a>, Electronic Colloquium on Computational Complexity 2011
%H A160389 Wolfdieter Lang, <a href="http://arxiv.org/abs/1210.1018">The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon</a>, arXiv:1210.1018 [math.GR], Oct 03 2012.
%H A160389 Peter Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden Fields: A Case for the Heptagon</a>, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
%H A160389 Ryan Williams, <a href="https://citeseerx.ist.psu.edu/pdf/54d8304f13676f7cbddd71d1d0f3801f243eed6e">Time-space tradeoffs for counting NP solutions modulo integers</a>, Computational Complexity 17 (2008), pp. 179-219.
%H A160389 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%F A160389 Equals 2*A073052. - _Michel Marcus_, Nov 21 2013
%F A160389 Equals (Re((-(4*7)*(1 + 3*sqrt(3)*i))^(1/3)) + 1)/3, with the real part Re, and i = sqrt(-1). - _Wolfdieter Lang_, Feb 24 2015
%F A160389 Equals i^(2/7) - i^(12/7). - _Peter Luschny_, Apr 04 2020
%F A160389 From _Peter Bala_, Oct 20 2021: (Start)
%F A160389 Equals 2 - (1 - z)*(1 - z^6)/((1 - z^3)*(1 - z^4)), where z = exp(2*Pi*i/7).
%F A160389 The other two zeros of the minimal polynomial x^3 - x^2 - 2*x + 1 of 2*cos(Pi/7) are given by 2 - (1 - z^3)*(1 - z^4)/((1 - z^2)*(1 - z^5)) = 2*cos(3*Pi/7) = A255241 and 2 - (1 - z^2)*(1 - z^5)/((1 - z)*(1 - z^6)) = cos(5*Pi/7) = -A362922.
%F A160389 Equals Product_{n >= 0} (7*n+2)*(7*n+5)/((7*n+1)*(7*n+6)) = 1 + Product_{n >= 0} (7*n+2)*(7*n+5)/((7*n+3)*(7*n+4)) = 1/A255240.
%F A160389 The linear fractional mapping r -> 1/(1 - r) cyclically permutes the three zeros of the minimal polynomial x^3 - x^2 - 2*x + 1. The inverse mapping is r -> (r - 1)/r.
%F A160389 The quadratic mapping r -> 2 - r^2 also cyclically permutes the three zeros. The inverse mapping is r -> r^2 - r - 1. (End)
%F A160389 Equals i^(2/7) + i^(-2/7). - _Gary W. Adamson_, Feb 11 2022
%F A160389 From _Amiram Eldar_, Nov 22 2024: (Start)
%F A160389 Equals Product_{k>=1} (1 - (-1)^k/A047336(k)).
%F A160389 Equals 1 + cosec(3*Pi/14)/2 = 1 + Product_{k>=1} (1 + (-1)^k/A047341(k)). (End)
%F A160389 Equals sqrt(A116425). - _Hugo Pfoertner_, Nov 22 2024
%e A160389 1.801937735804838252472204639014890102331838324263714300107124846398864...
%p A160389 evalf(2*cos(Pi/7), 100); # _Wesley Ivan Hurt_, Feb 01 2017
%t A160389 RealDigits[2 Cos[Pi/7], 10, 111][[1]] (* _Robert G. Wilson v_, Jun 11 2013 *)
%o A160389 (PARI)  default(realprecision, 20080); x=2*cos(Pi/7); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b160389.txt", n, " ", d));
%o A160389 (Magma) R:= RealField(200); Reverse(Intseq(Floor(10^110*2*Cos(Pi(R)/7)))); // _Marius A. Burtea_, Nov 13 2019
%Y A160389 Cf. A039921 (continued fraction).
%Y A160389 Cf. A047336, A047341, A116425, A255240, A255249.
%Y A160389 Cf. A003558 (the constant is cyclic with period 3, for N = 7).
%K A160389 nonn,cons,changed
%O A160389 1,2
%A A160389 _Harry J. Smith_, May 31 2009