A039921 -id:A039921 - cp's OEIS frontend

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

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

Offset: 1

Harry J. Smith, May 31 2009

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 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

			1.801937735804838252472204639014890102331838324263714300107124846398864...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.

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
Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv:1210.1018 [math.GR], Oct 03 2012.
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.
Index entries for algebraic numbers, degree 3

Cf. A039921 (continued fraction).
Cf. A047336, A047341, A116425, A255240, A255249.
Cf. A003558 (the constant is cyclic with period 3, for N = 7).

R:= RealField(200); Reverse(Intseq(Floor(10^110*2*Cos(Pi(R)/7)))); // Marius A. Burtea, Nov 13 2019

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

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

default(realprecision, 20080); x=2*cos(Pi/7); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b160389.txt", n, " ", d));

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
From Peter Bala, Oct 20 2021: (Start)
Equals 2 - (1 - z)*(1 - z^6)/((1 - z^3)*(1 - z^4)), where z = exp(2*Pi*i/7).
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.
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.
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.
The quadratic mapping r -> 2 - r^2 also cyclically permutes the three zeros. The inverse mapping is r -> r^2 - r - 1. (End)
Equals i^(2/7) + i^(-2/7). - Gary W. Adamson, Feb 11 2022
From Amiram Eldar, Nov 22 2024: (Start)
Equals Product_{k>=1} (1 - (-1)^k/A047336(k)).
Equals 1 + cosec(3*Pi/14)/2 = 1 + Product_{k>=1} (1 + (-1)^k/A047341(k)). (End)
Equals sqrt(A116425). - Hugo Pfoertner, Nov 22 2024

A316157 Positive integers Q such that there is a cubic x^3 - Qx + R that has three real roots whose continued fraction expansion have common tails.

Original entry on oeis.org

3, 7, 9, 21, 21, 39, 61, 63, 93, 129, 169, 171, 219, 273, 331, 333, 399, 471, 547, 549, 633, 723, 817, 819, 921, 1029, 1141, 1143, 1263, 1389, 1519, 1521, 1659, 1803, 1951, 1953, 2109, 2271, 2437, 2439, 2613, 2793, 2977, 2979, 3171, 3369, 3571, 3573, 3783, 3999, 4219, 4221, 4449, 4683, 4921, 4923

Offset: 1

Greg Dresden, Jun 25 2018

nonn

After 3, the prime terms appear to be the primes in A275878 (namely, 7, 61, 331, 547, 1951, ...)

			For the first entry of Q=3, we have the polynomial x^3 - 3x + 1. Its roots, expressed as continued fractions, all have a common tail of 3, 2, 3, 1, 1, 6, 11, ... The next examples are Q=7 with the polynomial x^3 - 7x + 7, then Q=9 with the polynomial x^3 - 9x + 9, and Q=21 with the polynomials x^3 - 21x + 35 and x^3 - 21x + 37. Note that for the Q=7 example, we get the common tail of 2, 3, 1, 6, 10, 5, ... which is contained in A039921.

Joseph-Alfred Serret, Section 512, Cours d'algèbre supérieure, Gauthier-Villars.

Cf. A316184. Contained in the union of A034017 and three times A034017.

SetOfQRs = {};
M = 1000;
Do[
  If[Divisible[3 (a^2 - a + 1), c^2] &&
    Divisible[(2 a - 1) (a^2 - a + 1), c^3] &&
    3 (a^2 - a + 1)/c^2 <=  M,
   SetOfQRs =
    Union[SetOfQRs, { { (3 (a^2 - a + 1))/
       c^2, ((2 a - 1) (a^2 - a + 1))/c^3}}   ]],
  {c, 1, M/3 + 1, 2}, {a, 1, Sqrt[M c^2/3 + 3/4] + 1/2}];
Print[SetOfQRs // MatrixForm];

More terms from Robert G. Wilson v, Jul 02 2018

A316184 Positive integers R such that there is a cubic x^3 - Qx + R that has three real roots whose continued fraction expansion have common tails.

Original entry on oeis.org

1, 7, 9, 35, 37, 91, 183, 189, 341, 559, 845, 855

Offset: 1

Greg Dresden, Jun 25 2018

			For the first entry of R=1, we have the polynomial x^3 - 3x + 1. Its roots, expressed as continued fractions, all have a common tail of 3, 2, 3, 1, 1, 6, 11, ... The next examples are R=7 with the polynomial x^3 - 7x + 7, then R=9 with the polynomial x^3 - 9x + 9, and Q=35 with the polynomial x^3 - 21x + 35. Note that for the R=7 example, we get the common tail of 2, 3, 1, 6, 10, 5, ... which is contained in A039921.

Joseph-Alfred Serret, Section 512, Cours d'algèbre supérieure, Gauthier-Villars.

Cf. A039921, A316157.

SetOfQRs = {};
M = 1000;
Do[
  If[Divisible[3 (a^2 - a + 1), c^2] &&
    Divisible[(2 a - 1) (a^2 - a + 1), c^3] &&
    3 (a^2 - a + 1)/c^2 <=  M,
   SetOfQRs =
    Union[SetOfQRs, { { (3 (a^2 - a + 1))/
       c^2, ((2 a - 1) (a^2 - a + 1))/c^3}}   ]],
  {c, 1, M/3 + 1, 2}, {a, 1, Sqrt[M c^2/3 + 3/4] + 1/2}];
Print[SetOfQRs // MatrixForm];

A072977 Increasing partial quotients of w = 2*cos(Pi/7).

Original entry on oeis.org

1, 4, 20, 39, 94, 636, 699, 716, 904, 1374, 1824, 2457, 24007, 32164, 170306, 179545, 198107, 463343, 579913, 910774, 3758763, 3896343, 5800335, 11314629, 13245450, 14422622, 62449915

Offset: 1

Alastair Rucklidge (A.M.Rucklidge(AT)leeds.ac.uk) and Robert G. Wilson v, Aug 13 2002

w satisfies w3 - w2 - 2w + 1 = 0 and so is algebraic.

			w = 1.801937735804838252472204639014890102331838324263714300...

Continued fraction: A039921.

a = ContinuedFraction[2*Cos[Pi/7], 10^6]; b = 0; Do[ If[a[[n]] > b, b = a[[n]]; Print[b]], {n, 1, 10^6 - 1}]

a(22)-a(27) from Robert G. Wilson v, Jun 12 2013.

A316389 Continued fraction expansion of largest root of x^3 - 7*x + 7.

Original entry on oeis.org

1, 1, 2, 4, 20, 2, 3, 1, 6, 10, 5, 2, 2, 1, 2, 2, 1, 18, 1, 1, 3, 2, 1, 2, 1, 2, 1, 39, 2, 1, 1, 1, 13, 1, 2, 1, 30, 1, 1, 1, 3, 2, 5, 4, 1, 5, 1, 5, 1, 2, 1, 1, 94, 6, 2, 19, 11, 1, 60, 1, 1, 50, 2, 1, 1, 8, 53, 1, 3, 1, 6, 3, 2, 1, 5, 1, 1, 3, 4, 636, 1, 2, 1, 3, 3, 7, 9, 1, 2, 10, 3, 1, 22, 1, 119, 3, 32, 1, 2, 1

Offset: 1

Greg Dresden, Jul 01 2018

a(n) is identical to A039921(n-1) for n >= 3. The largest root of x^3 - 7*x + 7 equals (3*w-1)/(2*w-1) for w = 2*cos(Pi/7), where w is the number referenced in A039921. Interestingly enough, all three roots of x^3-7*x+7 have a continued fraction expansion that ends in 2, 3, 1, 6, 10, 5, 2, 2, 1, ... which is a(n) for n >= 5.

			1.69202147163009586962781489700206914019726...

Cf. A039921.

ContinuedFraction[Root[x^3 - 7 x + 7, 3], 100]

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

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A316157 Positive integers Q such that there is a cubic x^3 - Qx + R that has three real roots whose continued fraction expansion have common tails.

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A316184 Positive integers R such that there is a cubic x^3 - Qx + R that has three real roots whose continued fraction expansion have common tails.

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A072977 Increasing partial quotients of w = 2*cos(Pi/7).

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A316389 Continued fraction expansion of largest root of x^3 - 7*x + 7.

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