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A183918 Characteristic sequence for cos(2*Pi/n) being rational. 6
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 (list; graph; refs; listen; history; text; internal format)
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
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.
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).
Sequence in context: A103368 A055132 A285403 * A128408 A121802 A156241
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jan 08 2011
STATUS
approved

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