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Search: id:A136571
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| A136571 |
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Irregular triangle of coefficients of the minimal polynomial of 2*cos(2*Pi/n) in decreasing powers. |
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+0
1
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| 1, -2, 1, 2, 1, 1, 1, 0, 1, 1, -1, 1, -1, 1, 1, -2, -1, 1, 0, -2, 1, 0, -3, 1, 1, -1, -1, 1, 1, -4, -3, 3, 1, 1, 0, -3, 1, 1, -5, -4, 6, 3, -1, 1, -1, -2, 1, 1, -1, -4, 4, 1, 1, 0, -4, 0, 2, 1, 1, -7, -6, 15, 10, -10, -4, 1, 1, 0, -3, -1, 1, 1, -8, -7, 21, 15, -20, -10, 5, 1, 1, 0, -5, 0, 5, 1, -1, -6, 6, 8, -8, 1, 1, -1, -4, 3, 3, -1, 1, 1, -10, -9, 36, 28
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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The degree of the n-th polynomial is A023022(n), the half-totient function for n>2. These polynomials are integral, monic, and irreducible over the integers. Hence 2*cos(2*Pi/n) is an algebraic integer. When n is prime, the n-th row is the same as the n-th row of A066170. Carlitz and Thomas give an algorithm for computing these polynomials.
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REFERENCES
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Scott Beslin and Valerio de Angelis, The minimal polynomials of sin(2 pi/p) and cos(2 pi/n), Math. Mag., 77 (2004), 146-149.
L. Carlitz and J. M. Thomas, Rational tabulated values of trigonometric functions, Amer. Math. Monthly, 69 (1962), 789-793.
Gregory P. Dresden, On the middle coefficient of a cyclotomic polynomial, Amer. Math. Monthly, 111 (2004), 531-533.
D. H. Lehmer, A note on trigonometric algebraic numbers, Amer. Math. Monthly, 40 (1933), 165-166.
William Watkins and Joel Zeitlin, The minimal polynomial of cos(2 pi/n), Amer. Math. Monthly, 100 (1993), 471-474.
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LINKS
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T. D. Noe, Rows n=1..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Trigonometry Angles
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EXAMPLE
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x-2, x+2, x+1, x, x^2+x-1, x-1, x^3+x^2-2x-1, x^2-2, x^3-3x+1, x^2-x-1
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MATHEMATICA
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Flatten[Table[Reverse[CoefficientList[MinimalPolynomial[2Cos[2Pi/n], x], x]], {n, 25}]]
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CROSSREFS
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Adjacent sequences: A136568 A136569 A136570 this_sequence A136572 A136573 A136574
Sequence in context: A029443 A078508 A029416 this_sequence A108149 A128583 A064391
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KEYWORD
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nice,sign,tabf
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Jan 07 2008
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