cp's OEIS Frontend

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 1-3 of 3 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

A231188 Coefficient table for the minimal polynomials of 2*sin(2*Pi/n). Rising powers of x.

Original entry on oeis.org

0, 1, 0, 1, -3, 0, 1, -2, 1, 5, 0, -5, 0, 1, -3, 0, 1, -7, 0, 14, 0, -7, 0, 1, -2, 0, 1, -3, 0, 9, 0, -6, 0, 1, 5, 0, -5, 0, 1, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, -1, 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, -4, 0, 1, 17, 0, -204, 0, 714, 0, -1122, 0, 935, 0, -442, 0, 119, 0, -17, 0, 1, -3, 0, 9, 0, -6, 0, 1
Offset: 1

Views

Author

Wolfdieter Lang, Nov 29 2013

Keywords

Comments

The length of row n is deg(n) + 1 = A093819(n) + 1, that is 2, 2, 3, 2, 5, 3, 7, 3, 7, 5, 11, 2, 13, 7, 9, 5, 17,...
See A181871 for the coefficient table for the integer but non-monic minimal polynomials of sin(2*Pi/n), n>=1, called there pi(n, x). The present minimal polynomials of 2*sin(2*Pi/n) are integer and monic, and they are given by
MP2sin2(n, x) = pi(n, x/2).

Examples

			The table a(n,m) starts:
---------------------------------------------------------------------------------
n\m   0   1    2  3    4  5     6  7    8  9    10  11   12  13   14 15 16 ...
1:    0   1
2:    0   1
3:   -3   0    1
4:   -2   1
5:    5   0   -5  0    1
6:   -3   0    1
7:   -7   0   14  0   -7  0     1
8:   -2   0    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:  -1   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   -4  0    1
17:  17   0 -204  0  714  0 -1122  0  935  0  -442   0  119   0  -17  0  1
...

Crossrefs

Cf. A093819, A181871, A181872/A181872, A232624.

Formula

a(n,m) = [x^m] MP2sin2(n, x), n>=1, m = 0, 1, ..., A093819(n), with the minimal polynomials of 2*sin(2*Pi/n), given above in a comment in terms of the ones for sin(2*Pi/n).

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.
Showing 1-3 of 3 results.

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