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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A093819 Algebraic degree of sin(2*Pi/n).

Original entry on oeis.org

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
Offset: 1

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Author

Eric W. Weisstein, Apr 16 2004

Keywords

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

Crossrefs

Cf. A055035, A023022 (alg. degree of cos(2*Pi/n)), A183919.
Cf. A181872/A181873, A181871, A231188. - Wolfdieter Lang, Nov 30 2013

Programs

  • 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 *)

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

A181871 Coefficient array for integer polynomial version of minimal polynomials of sin(2*Pi/n). Rising powers of x.

Original entry on oeis.org

0, 2, 0, 2, -3, 0, 4, -2, 2, 5, 0, -20, 0, 16, -3, 0, 4, -7, 0, 56, 0, -112, 0, 64, -2, 0, 4, -3, 0, 36, 0, -96, 0, 64, 5, 0, -20, 0, 16, -11, 0, 220, 0, -1232, 0, 2816, 0, -2816, 0, 1024, -1, 2, 13, 0, -364, 0, 2912, 0, -9984, 0, 16640, 0, -13312, 0, 4096, -7, 0, 56, 0, -112, 0, 64, 1, 0, -32, 0, 224, 0, -448, 0, 256, 2, 0, -16, 0, 16, 17, 0, -816, 0, 11424, 0, -71808, 0, 239360, 0, -452608, 0, 487424, 0, -278528, 0, 65536, -3, 0, 36, 0, -96, 0, 64
Offset: 1

Views

Author

Wolfdieter Lang, Jan 14 2011

Keywords

Comments

The sequence of row lengths of this array is A093819(n)+1: [2, 2, 3, 2, 5, 3, 7, 3, 7, 5, 11, ...].
pi(n,x) := Sum_{m=0..d(n)} a(n,m)*x^m, n >= 1, is related to the (monic) minimal polynomial of sin(2*Pi/n), called Pi(n,x), by pi(n,x) = (2^d(n))*Pi(n,x), with the degree sequence d(n)=A093819(n), and Pi(n,x) is given in A181872/A181873.
Pi(n,x)=Psi(c(n),x) with the minimal polynomials Psi(n,x) of cos(2*Pi/n), and c(n):=A178182(n).
The minimal polynomials of sin(2*Pi/n) are, e.g., treated in the Lehmer and Niven references. (Note the mistake in the Lehmer references explained in the W. Lang link.) The fundamental polynomials Psi(n,x) are also studied in the Watkins-Zeitlin reference, where a recurrence is given.
See A231188 for the (monic and integer) minimal polynomials of 2*sin(2*Pi/n). = Wolfdieter Lang, Nov 30 2013

Examples

			[0, 2], [0, 2], [-3, 0, 4], [-2, 2], [5, 0, -20, 0, 16], [-3, 0, 4], [-7, 0, 56, 0, -112, 0, 64], [-2, 0, 4], [-3, 0, 36, 0, -96, 0, 64], [5, 0, -20, 0, 16], ...
pi(2,x) = (2^1)*Pi(2,x) = 2*Psi(c(2),x) = 2*Psi(4,x) = 2*x.

References

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

Links

Crossrefs

Cf. A181877 (cos(2*Pi/n) case), A231188 (2*sin(2*Pi/n) case).

Programs

  • Mathematica
    ro[n_] := (cc = CoefficientList[ p = MinimalPolynomial[ Sin[2*(Pi/n)], x], x]; 2^Exponent[p, x]*(cc/Last[cc])); Flatten[ Table[ ro[n], {n, 1, 18}]] (* Jean-François Alcover, Sep 28 2011 *)

Formula

a(n,m) = [x^m]pi(n,x), n >= 1, m=0..A093819(n), and pi(n,x) defined above in the comments.

A231189 Coefficients of the algebraic number 2*sin(2*Pi/n) in the power basis of Q(2*cos(Pi/q(n))), with q(n) = A225975(n), n >= 1.

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 1, 0, 0, 0, 1, 0, -3, 0, 1, 0, 0, 0, 1, 0, 5, 0, -5, 0, 1, 0, -3, 0, 1, 0, -7, 0, 14, 0, -7, 0, 1, 0, 0, 1, 0, 9, 0, -30, 0, 27, 0, -9, 0, 1, 0, 0, 0, 5, 0, -5, 0, 1, 0, -7, 0, 22, 0, -13, 0, 2, 0, -3, 0, 1, 0, 13, 0, -91, 0, 182, 0, -156, 0, 65, 0, -13, 0, 1, 0, 0, 0, -4, 0, 5, 0, -1, 0, -15, 0, 140, 0, -378, 0, 450, 0, -275, 0, 90, 0, -15, 0, 1, 0, 0, -1, 1
Offset: 1

Views

Author

Wolfdieter Lang, Dec 04 2013

Keywords

Comments

The relevant trigonometric identity (used in the D. H. Lehmer and I. Niven references, given in A181871) is 2*sin(2*Pi/n) = 2*cos(2*Pi*(1/n -1/4)) = 2*cos(Pi*abs(n-4)/(2*n)) = 2*cos(Pi*p(n)/q(n)), with gcd(p(n), q(n)) = 1 (fraction p(n)/q(n) in lowest terms). One finds p(n) = A106609(n-4), n >=4, with p(1) = 3 , p(2) = 1 = p(3), and q(n) = A225975(n), n >= 1. See the comments on these two A-numbers. Therefore, 2*sin(2*Pi/n) = R(p(n), rho(q(n))), with rho(k) = 2*cos(Pi/k), and the R-polynomials (monic version of Chebyshev's T-polynomials) are given in A127672. It may happen that p(n), the degree of R, is >= delta(q(n)), the degree of the algebraic number rho(q(n)). Here delta(k) = A055034(k) is the degree of the minimal polynomial C(k, x) of rho(k) found under A187360. In this case one can reduce all rho(q(n)) powers >= delta(q(n)) with the help of the equation C(q(n), rho(q(n))) = 0. Thus the final result is 2*sin(2*Pi/n) = R(p(n), x) (mod C(q(n), x)) with x = rho(q(n)). Because R is an integer polynomial this shows that 2*sin(2*Pi/n) is an integer in the algebraic number field Q(rho(q(n))) of degree delta(q(n)).
The power basis of Q(rho(q(n))) is <1, rho(q(n)), ..., rho(q(n))^(delta(q(n))-1)>. Therefore the length of row n of this table is delta(q(n)).
The values n for which mod C(q(n), x) is in operation for the given formula for 2*sin(2*Pi/n) are those for which delta(q(n)) - p(n) <= 0, that is n = 1, 2, 12, 15, 18, 20, 21, 24, 25, 27, 28, 30,...
For the minimal polynomials of 2*sin(2*Pi/n) see the coefficient table A231188.

Examples

			[0], [0], [0, 1], [2], [0, 1, 0, 0], [0, 1], [0, -3, 0, 1, 0, 0], [0, 1], [0, 5, 0, -5, 0, 1], ...
The table a(n,m) begins (the trailing zeros are needed to have the correct degree for Q(rho(q(n)))):
n\m  0   1  2   3  4    5  6    7  8    9  10  11 12  13 14 15 16 17 ...
1:   0
2:   0
3:   0   1
4:   2
5:   0   1  0   0
6:   0   1
7:   0  -3  0   1  0    0
8:   0   1
9:   0   5  0  -5  0    1
10:  0  -3  0   1
11:  0  -7  0  14  0   -7  0    1  0    0
12:  1
13:  0   9  0 -30  0   27  0   -9  0    1   0   0
14:  0   5  0  -5  0    1
15:  0  -7  0  22  0  -13  0    2
16:  0  -3  0   1
17:  0  13  0 -91  0  182  0 -156  0   65   0 -13  0   1  0  0
18:  0  -4  0   5  0   -1
19:  0 -15  0 140  0 -378  0  450  0 -275   0  90  0 -15  0  1  0  0
20: -1   1
...
--------------------------------------------------------------------------
n=1:  2*sin(2*Pi/1) = 0. rho(q(1)) = rho(2) = 2*cos(Pi/2) = 0 and p(1) = 3. R(3, x) = -3*x + x^3 and C(2, x) = x. Therefore R(3, x) (mod C(2, x)) = 0. The degree of C(2, x) is delta(2) = A055034(2) = 1. Here one should use 1 for the undefined  rho(q(1))^0 in order to obtain a(1, 0) = 0.
n=2: 2*sin(2*Pi/2) = 0; rho(q(2)) = rho(2) =  0; p(2) = 1,  R(1, x) = x , C(2, x) = x and delta(2) = 1.  Therefore   R(1, x)  (mod C(1, x)) = 0.   Again, rho(2)^0 is put to 1 here, and a(2, 0) = 0.
n=5: 2*sin(2*Pi/5) = R(1, rho(10)) (mod C(10, rho(10)) =1* rho(10) (the degree of C(10,x) is delta(10) = 4, therefore the mod prescription is not needed).  Therefore, a(5, 0) =0, a(5,1) =1, a(n, m) = 0 for m=2, 3.
n =11: 2*sin(2*Pi/11) = R(7, x) (mod(C(22, x)) with x = rho(22), because p(11) = 7 and q(11) = 22. The degree of C(22, x) is delta(22) = 10, therefore the mod restriction is not needed and R(7, x) = -7*x + 14*x^3 - 7*x^5 + x^7. The coefficients produce the row [0, -7, 0, 14,  0,  -7, 0, 1, 0, 0] with the two trailing zeros needed to obtain the correct row length, namely delta(q(11)) = 10.

Crossrefs

Cf. A055034 (for delta), A106609 (for p), A225975 (for q), A127672 (for R), A187360 (for C), A181871, A231188.

Formula

a(n,m) = [x^m] (R(p(n), x) (mod C(q(n), x)), n >= 1, m = 0, 1, ..., delta(q(n)) - 1, where the R and C polynomials are found in A187360 and A127672, respectively. p(n) = A106609(n-4), n >=4, with p(1) = 3 , p(2) = 1 = p(3), and q(n) = A225975(n). Powers of x = rho(q(n)) = 2*cos(Pi/q(n)) appear in the table in increasing order.

A232630 Coefficient table for the minimal polynomials of 2*sin(4*Pi/n). Rising powers of x.

Original entry on oeis.org

0, 1, 0, 1, -3, 0, 1, 0, 1, 5, 0, -5, 0, 1, -3, 0, 1, -7, 0, 14, 0, -7, 0, 1, -2, 1, -3, 0, 9, 0, -6, 0, 1, 5, 0, -5, 0, 1, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, -3, 0, 1, 13, 0, -91, 0, 182, 0, -156, 0, 65, 0, -13, 0, 1, -7, 0, 14, 0, -7, 0, 1, 1, 0, -8, 0, 14, 0, -7, 0, 1, -2, 0, 1, 17, 0, -204, 0, 714, 0, -1122, 0, 935, 0, -442, 0, 119, 0, -17, 0, 1
Offset: 1

Views

Author

Wolfdieter Lang, Dec 17 2013

Keywords

Comments

The length of row n is A232626(n) + 1, that is 2, 2, 3, 2, 5, 3, 7, 2, 7, 5, 11, 3, 13, 7, 9, 3, 17, 7, 19, 5,...
In a regular n-gon, n>=2, inscribed in a circle of radius R (in some length units), 2*sin(4*Pi/n) = (S(n)/R)*(D(1,n)/S(n)) = D(1,n)/R, with the side length S(n) and the length of the first (smallest) diagonal D(1,n). For n=2 there is no such diagonal, and one can put D(1,2) = 0. Obviously, D(1,2*m) = S(m), m >= 2.
For the power basis representation of 2*sin(4*Pi/n) in the algebraic number field Q(rho(q(2,n))), with q(2,n)) = A232625(n) and rho(m) := 2*cos(Pi/m), see A232629. Call the row polynomials of A232629 PB2(n,x) (power basis polynomial for the case k=2 in 2*sin(2*Pi*k/n)).
The minimal polynomial of 2*sin(4*Pi/n), call it MP2(n, x), is obtained from the conjugates rho(q(2,n),j), j= 1, ... , delta(q(2,n)) = A232626(n), which are the zeros of C(q(2,n), x), the minimal polynomial of rho(q(2,n)) = rho(q(2,n),1) (for C see A187360). MP2(n, x) = product(x - PB2(n, rho(q(2,n),j)), j=1..A232626(n)) (mod C(q(2,n), rho(q(2,n)))).

Examples

			The table a(n,m) begins:
--------------------------------------------------------------------------------------
n\m   0  1    2  3     4  5     6  7      8  9   10 11   12 13   14 15   16 17 18 ...
1:    0  1
2:    0  1
3:   -3  0    1
4:    0  1
5:    5  0   -5  0     1
6:   -3  0    1
7:   -7  0   14  0    -7  0     1
8:   -2  1
9:   -3  0    9  0    -6  0     1
10:   5  0   -5  0     1
11: -11  0   55  0   -77  0    44  0    -11  0   1
12:  -3  0    1
13:  13  0  -91  0   182  0  -156  0     65  0 -13  0  1
14:  -7  0   14  0    -7  0     1
15:   1  0   -8  0    14  0    -7  0      1
16:  -2  0    1
17:  17  0 -204  0   714  0 -1122  0    935  0 -442  0  119  0  -17  0    1
18:  -3  0    9  0    -6  0     1
19: -19  0  285  0 -1254  0  2508  0  -2717  0 1729  0 -665  0  152  0  -19  0  1
20:   5  0   -5  0     1
...
n=1: 2*sin(4*Pi/1) = 0 is rational, therefore MP2(1, x) = x, with coefficients 0, 1, and degree A232626(1) = 1. PB2(1, rho(1,1)) = PB2(1, rho(1)) = 0.
n=3: A232626(2) = 2. PB2(2, x) = -x, C(6, x) = x^2 - 3, with zeros rho(6) and R(5, rho(6)) (for R see A127672), hence rho(6,1) = rho(6) and rho(6,2) = R(5, rho(6))=  5*rho(6) - 5*rho(6)^3 + 1*rho(6)^5, MP2(3, x) = (x - (-rho(6)))*(x - (- R(5, rho(6))) reduced with rho(6)^2 = 3 leading to MP2(3, x) = -3 + x^2, yielding row n=3: -3  0  1.
n=8: this row -2, 1 coincides with row n=4 of A231188.
n=17: coincides with WolframAlpha's MinimalPolynomial[2*sin(4*Pi/17),x] = 17-204 x^2+714 x^4-1122 x^6+935 x^8-442 x^10+119 x^12-17 x^14+x^16.

Crossrefs

Cf. A231188 (k=1 case), A187360 (C), A127672(R), A232626 (degree), A232629 (PB2).

Formula

a(n,m) = [x^m] MP2(n, x), n>=1, m = 0, 1, ..., A232626(n), with the minimal polynomials of 2*sin(4*Pi/n), computed like explained above in a comment.
a(2*l,m) = A231188(l,m), m = 0, 1, ..., A093819(l), l >= 1.

A232627 Discriminants of the minimal polynomials of 2*sin(2*Pi/n) for n >= 1.

Original entry on oeis.org

1, 1, 12, 1, 2000, 12, 1075648, 8, 1259712, 2000, 2414538435584, 1, 7340688973975552, 1075648, 324000000, 2048, 187591757103747287810048, 1259712, 1436650532447139184230793216, 5, 843466573910016, 2414538435584
Offset: 1

Views

Author

Wolfdieter Lang, Dec 12 2013

Keywords

Comments

The coefficient list for the minimal polynomials of 2*sin(2*Pi/n), called here MP(1; n, x), is given as A231188.

Examples

			n=5: MP(1; 5, x) = 5 - 5*x^2 + x^4 with the four zeros x[1] = +sqrt(2 + tau), x[2] = -sqrt(2 + tau), x[3] = +sqrt(3 - tau), x[4] = -sqrt(3 - tau), with the golden section tau := (1 + sqrt(5))/2. They produce the discriminant(MP(1; 5, x)) = (Det(Vandermonde(4,[x[1],x[2],x[3],x[4]])))^2 = (20*sqrt(5))^2 = 2000.

Links

Crossrefs

Cf. A231188, A093819.

Formula

a(n) = discriminant of MP(1; n, x) = sum(A231188(n,m)*x^m, m=0..deg(1; n)) with the degree deg(1; n) = A093819(n), n >= 1.
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