A039921 Continued fraction expansion of w = 2*cos(Pi/7).

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, 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

Offset: 0

Author

N. J. A. Sloane

Keywords

• cofr
• nonn

Comments

Arises in the approximation of 14-fold quasipatterns by 14 Fourier modes.

Examples

			w = 1.80193773580483825247220463901489010233183832426371430010712484639886...
Equals 1 + 1/(1 + 1/(4 + 1/(20 + 1/(2 + ...)))). - _Harry J. Smith_, May 31 2009
		

References

• A. M. Rucklidge & W. J. Rucklidge (preprint) 2002.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..20000
• S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134.
• S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134. [Annotated scanned copy]
• Alastair Rucklidge, Home page
• G. Xiao, Contfrac
• Index entries for continued fractions for constants

Crossrefs

Cf. A160389 (Decimal expansion). - Harry J. Smith, May 31 2009

Programs

Mathematica

ContinuedFraction[2*Cos[Pi/7], 100]

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

Formula

w satisfies w^3 - w^2 - 2w + 1 = 0 and so is algebraic.

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