

A160389


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


19



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, 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, 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
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OFFSET

1,2


COMMENTS

Arises in the approximation of 14fold quasipatterns by 14 Fourier modes.
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
rho(7):= 2*cos(Pi/7) is the length ratio (smallest diagonal)/side in the regular 7gon (heptagon). The algebraic number field Q(rho(7)) of degree 3 is fundamental for the 7gon. 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
An algebraic integer of degree 3 with minimal polynomial x^3  x^2  2x + 1.  Charles R Greathouse IV, Nov 12 2014
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
The constant is the square root of 3.24697... (cf. A116425). It's the fifth longest diagonal in the regular 14gon with unit radius, which equals 2*sin(5*Pi/14).  Gary W. Adamson, Nov 09 2019


LINKS

Harry J. Smith, Table of n, a(n) for n = 1..20000
Simon Baker, Exceptional digit frequencies and expansions in noninteger bases, arXiv:1711.10397 [math.DS], 2017. See Theorem 1.1 p. 3.
Sam Buss and Ryan Williams, Limits on alternationtrading proofs for timespace lower bounds, Electronic Colloquium on Computational Complexity 2011
Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular ngon, arXiv:1210.1018 [math.GR], Oct 3 2012.
Peter Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
Ryan Williams, Timespace tradeoffs for counting NP solutions modulo integers, Computational Complexity 17 (2008), pp. 179219.


FORMULA

Equals 2*A073052.  Michel Marcus, Nov 21 2013
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
Equals i^(2/7)  i^(12/7).  Peter Luschny, Apr 04 2020


EXAMPLE

1.801937735804838252472204639014890102331838324263714300107124846398864...


MAPLE

evalf(2*cos(Pi/7), 100); # Wesley Ivan Hurt, Feb 01 2017


MATHEMATICA

RealDigits[2 Cos[Pi/7], 10, 111][[1]] (* Robert G. Wilson v, Jun 11 2013 *)


PROG

(PARI) { default(realprecision, 20080); x=2*cos(Pi/7); for (n=1, 20000, d=floor(x); x=(xd)*10; write("b160389.txt", n, " ", d)); }
(MAGMA) R:= RealField(200); Reverse(Intseq(Floor(10^110*2*Cos(Pi(R)/7)))); // Marius A. Burtea, Nov 13 2019


CROSSREFS

Cf. A039921 Continued fraction.
Cf. A116425
Cf. A003558 (the constant is cyclic with period 3, for N = 7).
Sequence in context: A021559 A167176 A195447 * A011104 A232227 A322231
Adjacent sequences: A160386 A160387 A160388 * A160390 A160391 A160392


KEYWORD

nonn,cons


AUTHOR

Harry J. Smith, May 31 2009


STATUS

approved



