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 1-4 of 4 results.

A231190 Numerator of abs(n-8)/(2*n), n >= 1.

Original entry on oeis.org

7, 3, 5, 1, 3, 1, 1, 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 15, 31, 2, 33, 17, 35, 9, 37, 19, 39, 5, 41, 21, 43, 11, 45, 23, 47, 3, 49, 25, 51, 13, 53, 27, 55, 7, 57, 29, 59, 15, 61, 31, 63, 4, 65, 33, 67, 17
Offset: 1

Views

Author

Wolfdieter Lang, Dec 12 2013

Keywords

Comments

Because 2*sin(Pi*4/n) = 2*cos(Pi*abs(n-8)/(2*n)) = 2*cos(Pi*a(n)/b(n)) with gcd(a(n),b(n)) = 1, one has
2*sin(Pi*4/n) = R(a(n), x) (mod C(b(n), x)), with x = 2*cos(Pi/b(n)) =: rho(b(n)). The integer Chebyshev R and C polynomials are found in A127672 and A187360, respectively.
b(n) = A232625(n). This shows that 2*sin(Pi*4/n) is an integer in the algebraic number field Q(rho(b(n))) of degree delta(b(n)), with delta(k) = A055034(k). This degree delta(b(n)) is given in A231193(n), and if gcd(n,2) = 1 it coincides with the one for sin(2*Pi/n) given by A093819(n). See Theorem 3.9 of the I. Niven reference, pp. 37-38, which uses gcd(k, n) = 1. See also the Jan 09 2011 comment on A093819.
a(n) and b(n) = A232625(n) are the k=2 members of a family of pair of sequences p(k,n) and q(k,n), n >= 1, k >= 1, relevant to determine the algebraic degree of 2*sin(Pi*2*k/n) from the trigonometric identity (used in the D. H. Lehmer and I. Niven references) 2*sin(Pi*2*k/n) = 2*cos(Pi*abs(n-4*k)/(2*n)) = 2*cos(Pi*p(k,n)/q(k,n)). This is R(p(k,n), x) (mod C(q(k,n), x)), with x = 2*cos(Pi/q(k,n)) =: rho(q(k,n)). The polynomials R and C have been used above. C(q(k,n), x) is the minimal polynomial of rho(q(k,n)) with degree delta(q(k,n)), which is then the degree, call it deg(k,n), of the integer 2*sin(Pi*2*k/n) in the number field Q(rho(q(k,n))). From Theorem 3.9 of the I. Niven reference deg(k,n) is, for given k, for those n with gcd(k, n) = 1 determined by A093819(n). In general deg(k,n) = A093819(n/gcd(k,n)). For the k=1 instance p(1,n) and q(1,n) see comments on A106609 and A225975.

References

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

Links

Crossrefs

Cf. A127672 (R), A187360 (C), A232625 (b), A055034 (delta), A093819 (degree if k=1), A232626(degree if k=2), A106609 (k=1, p), A225975 (k=1, q), A106617.

Programs

  • Maple
    f:= n -> numer(abs(n-8)/(2*n)):
map(f, [$1..100]); # Robert Israel, Dec 06 2018
  • Mathematica
    a[n_] := Numerator[Abs[n-8]/(2n)]; Array[a, 50] (* Amiram Eldar, Dec 06 2018 *)

Formula

a(n) = numerator(abs(n-8)/(2*n)), n >= 1.
a(n) = abs(n-8)/gcd(n-8, 16).
a(n) = abs(n-8) if n is odd; if n is even then a(n) = abs(n-8)/2 if n/2 == 1, 3, 5, 7 (mod 8), a(n) = abs(n-8)/4 if n/2 == 2, 6 (mod 8), a(n) = abs(n-8)/8 if n/2 == 0 (mod 8) and a(n) = abs(n-8)/16 if n == 4 (mod 8).
O.g.f.: 1+ x*(7 + 3*x + 5*x^2 + 1*x^3 + 3*x^4 + 1*x^5 + 1*x^6) + N(x)/(1-x^16)^2 , with N(x) = x^9*((1+x^30) + x*(1+x^28) + 3*x^2*(1+x^26) + x^3*(1+x^24) + 5*x^4*(1+x^22) + 3*x^5*(1+x^20) + 7*x^6*(1+x^18) + x^7*(1+x^16) + 9*x^8*(1+x^14) + 5*x^9*(1+x^12) + 11*x^10*(1+x^10) + 3*x^11*(1+x^8) + 13*x^12*(1+x^6) + 7*x^13*(1+x^4) + 15*x^14*(1+x^2)+x^15).
a(n+32)-2*a(n+16)+a(n) = 0 for n >= 8.
a(n+8) = A106617(n). - Peter Bala, Feb 28 2019

A232625 Denominators of abs(n-8)/(2*n), n >= 1.

Original entry on oeis.org

2, 2, 6, 2, 10, 6, 14, 1, 18, 10, 22, 6, 26, 14, 30, 4, 34, 18, 38, 10, 42, 22, 46, 3, 50, 26, 54, 14, 58, 30, 62, 8, 66, 34, 70, 18, 74, 38, 78, 5, 82, 42, 86, 22, 90, 46, 94, 12, 98, 50, 102, 26, 106, 54, 110, 7, 114, 58, 118, 30, 122, 62, 126, 16, 130, 66
Offset: 1

Views

Author

Wolfdieter Lang, Dec 12 2013

Keywords

Comments

The numerators are given in A231190. See the comments there on 2*sin(Pi*4/n).
2*sin(Pi*4/n) = R(b(n), x) (mod C(b(n), x)), with x = 2*cos(Pi/a(n)) =: rho(a(n)). The integer Chebyshev R and C polynomials are found in A127672 and A187360, respectively. b(n) = A231190(n).
delta(a(n)) = deg(2,n), with delta(k) = A055034(k), is the degree of the algebraic number 2*sin(Pi*4/n) given in A232626.

Links

Crossrefs

Cf. A127672 (R), A187360 (C), A231190 (b), A055034 (delta), A232626 (degree k=2), A106609 (k=1, p), A225975 (k=1, q), A093819 (degree k=1).

Programs

  • Mathematica
    a[n_] := Denominator[(n-8)/(2*n)]; Array[a, 100] (* Amiram Eldar, Nov 09 2024 *)
  • PARI
    a(n) = denominator((n-8)/(2*n)); \\ Amiram Eldar, Nov 09 2024

Formula

a(n) = denominator(abs(n-8)/(2*n)), n >= 1.
a(n) = 2*n/gcd(n-8, 16).
a(n) = 2*n if n is odd; if n is even then a(n) = n if n/2 == 1, 3, 5, 7 (mod 8), a(n) = n/2 if n/2 == 2, 6 (mod 8), a(n) == n/4 if n/2 == 0 (mod 8) and a(n) = n/8 if n == 4 (mod 8).
O.g.f.: x*(2*(1+x^30) + 2*x*(1+x^28) + 6*x^2*(1+x^26) + 2*x^3*(1+x^24) + 10*x^4*(1+x^22) + 6*x^5*(1+x^20) + 14*x^6*(1+x^18) + x^7*(1+x^16) + 18*x^8*(1+x^14) + 10*x^9*(1+x^12) + 22*x^10*(1+x^10) + 6*x^11*(1+x^8) + 26*x^12*(1+x^6) + 14*x^13*(1+x^4) + 30*x^14*(1+x^2) + 4*x^15)/(1-x^16)^2.
Sum_{k=1..n} a(k) ~ (171/256) * n^2. - Amiram Eldar, Nov 09 2024

A232629 Coefficients of the algebraic number 2*sin(4*Pi/n) in the power basis of Q(2*cos(Pi/q(2,n))), with q(2,n) = A232625(n), n >= 1.

Original entry on oeis.org

0, 0, 0, -1, 0, 0, -3, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, -3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 5, 0, -5, 0, 1, 0, 0, 0, 0, 0, 0, 0, -3, 0, 1, 0, 0, 0, -7, 0, 14, 0, -7, 0, 1, 0, 1, 0, 9, 0, -30, 0, 27, 0, -9, 0, 1, 0, 0, 0, 0, 0, 0, 0, 5, 0, -5, 0, 1, 0, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, 0, 0, 0, 0, 0, 0, 0, -3, 0, 1
Offset: 1

Views

Author

Wolfdieter Lang, Dec 17 2013

Keywords

Comments

The length of row n is A232626(n).
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.
See a comment on A231190 regarding the pair of sequences p(k,n) and q(k,n), n >= 1, k >= 1. Here k=2 with A231190 and A232625.
See also the k=1 analog A231189 of the present table.
The relevant identity is here 2*sin(Pi*4/n) = 2*cos(Pi*abs(n-8)/(2*n)) = 2*cos(Pi*p(2,n)/q(2,n)). This is R(p(2,n), x) (mod C(q(2,n), x)), with x = 2*cos(Pi/q(2,n)) =: rho(q(2,n)). with the coefficient tables for the polynomials R and C given in A127672 and A187360, respectively. This gives the power base coefficients of 2*sin(Pi*4/n) in the algebraic number field Q(rho(q(2,n))) of degree delta(q(2,n)), with delta(n) = A055034(n), shown in A232626.
If the degree p(2,n) of R(p(2,n), x) is smaller than the degree A232626(n) of C(q(2,n), x) then 2*sin(Pi*4/n) = R(p(2,n), x). Otherwise the (mod C(q(2,n), x)) congruence is needed. This happens for n = 1, 2, 3, 4, 21, 24, 27, 30,...
The power basis of Q(rho(q(2,n))) is <1, rho(q(2,n)), ..., rho(q(2,n))^(delta(q(2,n))-1)>. Therefore the length of row n of this table is delta(q(2,n)) = A232626(n).
The coefficient table for the minimal polynomial of 2*sin(Pi*4/n) is given in A232630.

Examples

			The table a(n,m) begins (the trailing zeros are needed to have the correct degree A232626(n) in Q(rho(q(2,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:  0
5:  0  -3  0   1
6:  0   1
7:  0   1  0   0  0   0
8:  2
9:  0   1  0   0  0   0
10: 0   1  0   0
11: 0  -3  0   1  0   0  0   0  0   0
12: 0   1
13: 0   5  0  -5  0   1  0   0  0   0  0  0
14: 0  -3  0   1  0   0
15: 0  -7  0  14  0  -7  0   1
16: 0   1
17: 0   9  0 -30  0  27  0  -9  0   1  0  0  0  0  0  0
18: 0   5  0  -5  0   1
19: 0 -11  0  55  0 -77  0  44  0 -11  0  1  0  0  0  0  0  0
20: 0  -3  0   1
...
n=1: 2*sin(Pi*4/1) = 0. R(p(2,1), x) = R(7, x) = -7*x + 14*x^3 -7*x^5 + x^7. C(q(2,1), x) = C(2, x) = x, hence R(7, x) (mod C(2, x)) == 0, and
  with A232626(1) = 1, a(1,0) = 0.n=7: p(2,7) = A231190(7) = 1, q(2,7) = A232625(7) = 14. 2*sin(Pi*4/7) = R(1, x) = x = rho(14) := 2*cos(Pi/14). C(14, x) of degree 6 does not apply here. A232626(7) = 6, hence the row n=7 is  0  1  0  0  0  0.
n=9: p(2,9) = 1, q(2,9) = 18. 2*sin(Pi*4/9) = R(1, x) = x = rho(18) = 2*cos(Pi/18). C(18, x) with degree 6 is not needed here. A232626(9) = 6, hence row n=9 is also 0  1  0  0  0  0.
n=8: this row with entry 2 coincides with row n=4 of A231189.
n=17: row length A232626(17) = 16; p(2,17) = 9; C(34, x) has degree 16, therefore the R(9, x) coefficients produce here the first 10 entries for row n=17: 0  9  0 -30  0  27  0 -9  0  1, followed by 6 zeros, and 2*sin(Pi*4/17) = 9*rho(34) - 30*rho(34)^3 + 27*rho(34)^5 - 9*rho(34)^7 + 1*rho(34)^9, with rho(34) = 2*cos(Pi/34).

Crossrefs

Cf. A231190 (p), A232625 (q), A127672 (R), A187360 (C), A232626 (degree), A231189 (k=1 case), A232630 (minimal polynomials).

Formula

a(n,m) = [x^m] (R(p(2,n), x) (mod C(q(2,n), x)), n >= 1, m = 0, 1, ..., A232626(n) - 1, where the C and R polynomials are found in A187360 and A127672, respectively. p(2,n) = A231190(n) and q(2,n) = A232625(n). Powers of x = rho(q(2,n)) := 2*cos(Pi/q(2,n)) appear in the table in increasing order.
a(2*l,m) = A231189(l,m), l >= 1, m = 0, 1, ..., (A093819(n)-1).

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.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.