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Characteristic sequence for cos(2*Pi/n) being rational.
6

%I #53 Nov 08 2022 08:05:49

%S 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,

%T 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,

%U 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

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

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

%C 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.

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

%C Binary expansion of 61/64. - _Moritz Firsching_, Mar 01 2016

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

%H Wolfdieter Lang, <a href="/A181875/a181875.pdf">A181875/A181876. Minimal polynomials of cos(2Pi/n).</a>

%H D. H. Lehmer, <a href="http://www.jstor.org/stable/2301023">A Note on Trigonometric Algebraic Numbers</a>, Am. Math. Monthly 40 (3) (1933) 165-6.

%H W. Watkins and J. Zeitlin, <a href="http://www.jstor.org/stable/2324301">The Minimal Polynomial of cos(2Pi/n)</a>, Am. Math. Monthly 100,5 (1993) 471-4.

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

%F 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.

%F 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.

%e 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.

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

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

%Y Cf. A000010, A181875, A181876, A181877.

%K nonn,easy

%O 1

%A _Wolfdieter Lang_, Jan 08 2011