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A181877
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Integer coefficient array for polynomials related to the minimal polynomials of cos(2Pi/n). Rising powers of x.
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9
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-2, 2, 2, 2, 1, 2, 0, 2, -1, 2, 4, -1, 2, -1, -4, 4, 8, -2, 0, 4, 1, -6, 0, 8, -1, -2, 4, 1, 6, -12, -32, 16, 32, -3, 0, 4, -1, 6, 24, -32, -80, 32, 64, 1, -4, -4, 8, 1, 8, -16, -8, 16, 2, 0, -16, 0, 16, 1, -8, -40, 80, 240, -192, -448, 128, 256, -1, -6, 0, 8, 1, 10, -40, -160, 240, 672, -448, -1024, 256, 512, 5, 0, -20, 0, 16, 1, -16, 32, 48, -96, -32, 64, -1, 6, 12, -32, -16, 32, -1, -12, 60, 280, -560, -1792, 1792, 4608, -2304, -5120, 1024, 2048, 1, 0, -16, 0, 16, -1, 10, 100, -40, -800, 32, 2240, 0, -2560, 0, 1024, -1, -6, 24, 32, -80, -32, 64, 1, 18, 0, -240, 0, 864, 0, -1152, 0, 512, -7, 0, 56, 0, -112, 0, 64, -1, 14, 112, -448, -2016, 4032, 13440, -15360, -42240, 28160, 67584, -24576, -53248, 8192, 16384, 1, -8, -16, 8, 16
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OFFSET
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1,1
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COMMENTS
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The sequence of row lengths is d(n)+1, with d(n):=A023022(n), n>=2, and d(1):=1: [2, 2, 2, 2, 3, 2, 4, 3, 4, 3, 6, 3, 7, 4, 5, 5, 9, 4, 10, 5, 7,...].
psi(n,x):=sum(a(n,m)*x^m,m=0..d(n)), with the degree d(n):=A023022(n), n>=2, d(1):=1, equals (2^d(n))*Psi(n,x), with the minimal polynomials Psi(n,x) of cos(2*Pi/n), n>=1. See A181875/A181876 for the rational coefficient array of the monic Psi(n,x).
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REFERENCES
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D. H. Lehmer, A Note on Trigonometric Algebraic Numbers, Am. Math. Monthly 40,3 (1933) 165-6.
I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.
W. Watkins and J. Zeitlin, The Minimal Polynomial of cos(2Pi/n), Am. Math. Monthly 100,5 (1993) 471-4.
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LINKS
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Table of n, a(n) for n=1..170.
W. Lang: A181875/A181876. Minimal polynomials for cos(2Pi/n).
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FORMULA
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a(n,m) = [x^m]((2^d(n))*Psi(n,x)), with the minimal polynomials Psi(n,x) of cos(2*Pi/n), n>=1. See A18175(n,m)/A181876(n,m) for the rational Psi(n,x) coefficients.
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EXAMPLE
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[-2, 2], [2, 2], [1, 2], [0, 2], [-1, 2, 4], [-1, 2], [-1, -4, 4, 8], [-2, 0, 4], [1, -6, 0, 8], [-1, -2, 4], [1, 6, -12, -32, 16, 32],...
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MATHEMATICA
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ro[n_] := (cc = CoefficientList[ p = MinimalPolynomial[ Cos[2*(Pi/n)], x], x]; 2^Exponent[p, x]*(cc / Last[cc])); Flatten[ Table[ ro[n], {n, 1, 30}]] (* From Jean-François Alcover, Sep 28 2011 *)
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CROSSREFS
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Cf. A023022, A181875, A181876, A183918.
Sequence in context: A037200 A180174 A172363 * A175357 A090044 A036238
Adjacent sequences: A181874 A181875 A181876 * A181878 A181879 A181880
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KEYWORD
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sign,easy,tabf
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AUTHOR
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Wolfdieter Lang, Jan 08 2011
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STATUS
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approved
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