|
%I #24 Jul 03 2018 05:16:12
%S 1,1,4,20,2,3,1,6,10,5,2,2,1,2,2,1,18,1,1,3,2,1,2,1,2,1,39,2,1,1,1,13,
%T 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,
%U 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
%N Continued fraction expansion of w = 2*cos(Pi/7).
%C Arises in the approximation of 14-fold quasipatterns by 14 Fourier modes.
%D A. M. Rucklidge & W. J. Rucklidge (preprint) 2002.
%H Harry J. Smith, <a href="/A039921/b039921.txt">Table of n, a(n) for n = 0..20000</a>
%H S. Lang and H. Trotter, <a href="http://dx.doi.org/10.1515/crll.1972.255.112">Continued fractions for some algebraic numbers</a>, J. Reine Angew. Math. 255 (1972), 112-134.
%H S. Lang and H. Trotter, <a href="/A002945/a002945.pdf">Continued fractions for some algebraic numbers</a>, J. Reine Angew. Math. 255 (1972), 112-134. [Annotated scanned copy]
%H Alastair Rucklidge, <a href="http://www.maths.leeds.ac.uk/~alastair/">Home page</a>
%H G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H <a href="/index/Con#confC">Index entries for continued fractions for constants</a>
%F w satisfies w^3 - w^2 - 2w + 1 = 0 and so is algebraic.
%F The other two roots are 2*cos(3 Pi/7) and 2*cos(5 Pi/7); their continued fraction expansions also end in 20, 2, 3, 1, 6, 10, 5, 2, 2, 1, ... which is a(n) for n >= 3. - _Greg Dresden_, Jul 01 2018
%e w = 1.80193773580483825247220463901489010233183832426371430010712484639886...
%e Equals 1 + 1/(1 + 1/(4 + 1/(20 + 1/(2 + ...)))). - _Harry J. Smith_, May 31 2009
%t ContinuedFraction[2*Cos[Pi/7], 100]
%o (PARI) { allocatemem(932245000); default(realprecision, 21000); x=contfrac(2*cos(Pi/7)); for (n=0, 20000, write("b039921.txt", n, " ", x[n+1])); } \\ _Harry J. Smith_, May 31 2009
%Y Cf. A160389 (Decimal expansion). - _Harry J. Smith_, May 31 2009
%K cofr,nonn
%O 0,3
%A _N. J. A. Sloane_
|
