A183918 Characteristic sequence for cos(2*Pi/n) being rational.

1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

1

Comments

Sequence 1, 1, 1, 0, 1, followed by zeros.

The minimal polynomial of cos(2*Pi/n) has degree 1 iff a(n)=1. See, e.g., the Niven reference for the definition of minimal polynomial of an algebraic number on p. 28, the Corollary 3.12 on p. 41, and one of the tables in the D. H. Lehmer reference, p. 166.

In the Watkins and Zeitlin reference a recurrence for the minimal polynomial of cos(2*Pi/n) is found.

Binary expansion of 61/64. - Moritz Firsching, Mar 01 2016

References

I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.

Links

Table of n, a(n) for n=1..127.

Wolfdieter Lang, A181875/A181876. Minimal polynomials of cos(2Pi/n).

D. H. Lehmer, A Note on Trigonometric Algebraic Numbers, Am. Math. Monthly 40 (3) (1933) 165-6.

W. Watkins and J. Zeitlin, The Minimal Polynomial of cos(2Pi/n), Am. Math. Monthly 100,5 (1993) 471-4.

Index entries for characteristic functions

Formula

a(n)=1 if cos(2*Pi/n) is rational, and a(n)=0 if it is irrational. The rational values for n = 1, 2, 3, 4, 6, are 1, -1, -1/2, 0, +1/2, respectively.

a(n)=1 if Psi(n,x), the characteristic polynomial of cos(2*Pi/n), has degree 1, and a(n)=0 otherwise. See the Watkins and Zeitlin reference for Psi(n,x), called there Psi_n(x). See also the comment by A. Jasinski on A023022, and the W. Lang link for a table for n = 1..30.

Example

Psi(6,x) = x - 1/2 and Psi(5,x) = x^2 - (1/2)*x - 1/4. Therefore a(6)=1 and a(5)=0.

Crossrefs

Cf. A023022 (the degree sequence with A023022(1):=1).

Cf. A183919 (the characteristic sequence for sin(2*Pi/n) being rational).

Cf. A000010, A181875, A181876, A181877.

Sequence in context: A103368 A055132 A285403 * A128408 A121802 A156241

Adjacent sequences: A183915 A183916 A183917 * A183919 A183920 A183921

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 08 2011

Status

approved