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Search: a183919 -id:a183919
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A093819
Algebraic degree of sin(2*Pi/n).
+10
14
1, 1, 2, 1, 4, 2, 6, 2, 6, 4, 10, 1, 12, 6, 8, 4, 16, 6, 18, 2, 12, 10, 22, 4, 20, 12, 18, 3, 28, 8, 30, 8, 20, 16, 24, 3, 36, 18, 24, 8, 40, 12, 42, 5, 24, 22, 46, 8, 42, 20, 32, 6, 52, 18, 40, 12, 36, 28, 58, 4, 60, 30, 36, 16, 48, 20, 66, 8, 44, 24, 70, 12, 72, 36, 40, 9, 60, 24
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
The degree formula given in the I. Niven reference on p. 37-8 (see below) appears as part of theorem 3.9 attributed to D. H. Lehmer. However, this part, concerning sin(2*Pi/n), differs from Lehmer's result, which in fact is incorrect. - Wolfdieter Lang, Jan 09 2011
This is also the algebraic degree of the area of a regular n-gon inscribed in the unit circle. - Jack W Grahl, Jan 10 2011
Every degree appears in this sequence except for the half-nontotients, A079695. - T. D. Noe, Jan 12 2011
See A181872/A181873 for the monic rational minimal polynomial of sin(2*Pi/n), and A181871 for the non-monic integer version. In A231188 the (monic and integer) minimal polynomials for 2*sin(2*Pi/n) are given. - Wolfdieter Lang, Nov 30 2013
REFERENCES
I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
Sameen Ahmed Khan, Trigonometric Ratios Using Algebraic Methods, Mathematics and Statistics (2021) Vol. 9, No. 6, 899-907.
Eric Weisstein's World of Mathematics, Trigonometry Angles
FORMULA
a(4)=1, a(n)=phi(n) if gcd(n,8)<4; a(n)=phi(n)/4 if gcd(n,8)=4, and a(n)=phi(n)/2 if gcd(n,8)>4. Here phi(n)=A000010(n) (Euler totient). See the I. Niven reference, Theorem 3.9, p. 37-8. - Wolfdieter Lang, Jan 09 2011
a(n) = delta(c(n)/2) if c(n) = A178182(n) is even, and delta(c(n)) if c(n) is odd, with delta(n) = A055034(n), the degree of the algebraic number 2*cos(Pi/n). - Wolfdieter Lang, Nov 30 2013
MATHEMATICA
a[4]=1; a[n_] := Module[{g=GCD[n, 8], e=EulerPhi[n]}, If[g<4, e, If[g==4, e/4, e/2]]]; Array[a, 1000]
f[n_] := Exponent[ MinimalPolynomial[ Sin[ 2Pi/n]][x], x]; Array[f, 75] (* Robert G. Wilson v, Jul 28 2014 *)
CROSSREFS
Cf. A055035, A023022 (alg. degree of cos(2*Pi/n)), A183919.
Cf. A181872/A181873, A181871, A231188. - Wolfdieter Lang, Nov 30 2013
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Apr 16 2004
STATUS
approved
A183918
Characteristic sequence for cos(2*Pi/n) being rational.
+10
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
Table of n, a(n) for n=1..127.
Wolfdieter Lang, A181875/A181876. Minimal polynomials of cos(2Pi/n).
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.
Index entries for characteristic functions
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).
Cf. A000010, A181875, A181876, A181877.
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jan 08 2011
STATUS
approved
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