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 A160389 Decimal expansion of 2*cos(Pi/7). 18
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Arises in the approximation of 14-fold 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 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 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 14-gon 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 non-integer bases, arXiv:1711.10397 [math.DS], 2017. See Theorem 1.1 p. 3. Sam Buss and Ryan Williams, Limits on alternation-trading proofs for time-space lower bounds, Electronic Colloquium on Computational Complexity 2011 Peter Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997. Ryan Williams, Time-space tradeoffs for counting NP solutions modulo integers, Computational Complexity 17 (2008), pp. 179-219. 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][] (* 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=(x-d)*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

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Last modified July 6 02:35 EDT 2020. Contains 335475 sequences. (Running on oeis4.)