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A178182
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Minimal polynomials of sin(2Pi/n) are mapped to those of cos(2Pi/a(n)).
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5
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4, 4, 12, 1, 20, 12, 28, 8, 36, 20, 44, 6, 52, 28, 60, 16, 68, 36, 76, 5, 84, 44, 92, 24, 100, 52, 108, 14, 116, 60, 124, 32, 132, 68, 140, 9, 148, 76, 156, 40, 164, 84, 172, 22, 180, 92, 188, 48, 196, 100, 204, 13, 212, 108, 220, 56, 228, 116, 236, 30, 244, 124, 252, 64, 260, 132, 268, 17, 276, 140, 284, 72, 292, 148, 300, 38, 308, 156, 316, 80
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OFFSET
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1,1
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COMMENTS
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The minimal polynomials of cos(2*Pi/n) are treated, e.g. in the Lehmer, Niven and Watkins-Zeitlin references. Lehmer and Niven call them psi_n(x) (eq. (1) and Lemma 3.8, p.37, respectively). In the latter reference they are called Psi_n(x), and we call them Psi(n,x). By definition (Niven, p. 28) these are monic, rational polynomials which have as a root cos(2*Pi/n) and are of minimal degree. They are irreducible (Niven p. 37, Lemma 3.8). See also A181875 for more details and a link with Psi(n,x), n=1..30.
The minimal polynomials of sin(2*Pi/n) are treated, e.g. in the Lehmer and Niven references. Lehmer's theorem 2 is, however, incorrect. See A181872 and the link there for a counterexample. In this link one can also find these polynomials, called Pi(n,x), for n=1..30.
The sequence a(n) translates these polynomials: Pi(n,x)=Psi(a(n),x), n>=1. This translation is based on the trigonometric identity: sin(2*Pi/n)=cos(2*Pi*r(n)), with r(n):=|(4-n)/(4*n)|.
a(n):=denominator(r(n)) (in lowest terms). Note that the degrees agree with those given in the Niven reference, Theorem 3.9, p. 37.
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REFERENCES
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I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.
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LINKS
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Michael De Vlieger, Table of n, a(n) for n = 1..10000
D. H. Lehmer, A Note on Trigonometric Algebraic Numbers, Am. Math. Monthly 40,3 (1933) 165-6.
Pinthira Tangsupphathawat, Takao Komatsu, Vichian Laohakosol, Minimal Polynomials of Algebraic Cosine Values, II, J. Int. Seq., Vol. 21 (2018), Article 18.9.5.
W. Watkins and J. Zeitlin, The Minimal Polynomial of cos(2Pi/n), Am. Math. Monthly 100,5 (1993) 471-4.
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FORMULA
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a(n) = denominator(|(n-4)/(4*n)|), n>=1.
a(n) = 4*n/gcd(n-4,16). a(n) = 4*n if n is odd; if n is even then a(n) = 2*n if n/2 == 1, 3, 5, 7 (mod 8), a(n) = n if n/2 == 0, 4 (mod 8), a(n) = n/2 if n/2 == 6 (mod 8) and a(n) = n/4 if n/2 == 2 (mod 8). - Wolfdieter Lang, Dec 01 2013
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EXAMPLE
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Pi(5,x)=Psi(20,x) because sin(2*Pi/5)=cos(2*Pi/20).
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MATHEMATICA
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Array[4 #/GCD[# - 4, 16] &, 80] (* Michael De Vlieger, Feb 07 2019 *)
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CROSSREFS
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Cf. A181872.
Sequence in context: A014012 A273391 A273452 * A160721 A151836 A147582
Adjacent sequences: A178179 A178180 A178181 * A178183 A178184 A178185
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Jan 11 2011
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STATUS
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approved
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