A231190 -id:A231190 - cp's OEIS frontend

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

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

Wolfdieter Lang, Dec 12 2013

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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

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).

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

a(n) = denominator((n-8)/(2*n)); \\ Amiram Eldar, Nov 09 2024

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

A232626 Degree of the algebraic number 2sin(4Pi/n).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 6, 4, 10, 2, 12, 6, 8, 2, 16, 6, 18, 4, 12, 10, 22, 1, 20, 12, 18, 6, 28, 8, 30, 4, 20, 16, 24, 6, 36, 18, 24, 2, 40, 12, 42, 10, 24, 22, 46, 4, 42, 20, 32, 12, 52, 18, 40, 3, 36, 28, 58, 8, 60, 30, 36, 8, 48, 20, 66, 16, 44, 24, 70, 3, 72, 36, 40

Offset: 1

Wolfdieter Lang, Dec 12 2013

See the comment on A231190 for the formula for 2*sin(Pi*4/n) = 2*cos(Pi*p(2,n)/q(2,n)) with gcd(p(2,n),q(2,n)) = 1, where p(2,n) = A231190(n) and q(2,n) = A232625(n). This shows that 2*sin(Pi*4/n) is an integer in the algebraic number field Q(rho(q(2,n))) of degree a(n) = delta(q(2,n)) with delta(k) = A055034(k).

This degree a(n) is given by I. Niven's Theorem 3.9, pp. 37-38, by Niven(n/gcd(2,n)) with Niven(n) = A093819(n) the degree of 2*sin(2*Pi/n). Note that Niven uses gcd(k, n) = 1 in the derivation, and Niven(4) = 1. See the bisection given in the formula section which is obtained from this.

			a(1) = A093819(1) = 1; a(4) = phi(2) = 1; a(6) = phi(3) = 2; a(8) = 1; a(9) = A093819(9) = 6.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..175 from Vincenzo Librandi)

Cf. A000010, A055034, A231190, A232625, A093819.

f[n_] := Exponent[ MinimalPolynomial[ 2Sin[ 4Pi/n]][x], x]; Array[f, 75] (* Robert G. Wilson v, Jul 28 2014 *)

a(n) = {my(k = denominator((n-8)/(2*n))); if(k == 1, 1, eulerphi(2*k)/2);} \\ Amiram Eldar, Nov 09 2024

a(n) = delta(A232625(n)), n >=1, with delta(1) = 1 and delta(k) = phi(2*k)/2 with Euler's totient function phi (A000010). delta(k) = A055034(k).
a(2*k+1) = A093819(2*k+1), k >= 0.
For k >= 1: a(2*k) = A093819(k), that is a(2*k) = 1 if k=4, phi(k) if k odd or k == 2 (mod 4), phi(k)/2 if k == 0 (mod 8), phi(k)/4 if k == 4 (mod 8) (but not k=4).

A367824 Array read by ascending antidiagonals: A(n, k) is the numerator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

Original entry on oeis.org

1, 1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 1, 0, -1, -1, 1, 3, 1, -1, -3, -1, 1, 2, 1, 0, -1, -2, -1, 1, 5, 3, 1, -1, -3, -5, -1, 1, 3, 1, 1, 0, -1, -1, -3, -1, 1, 7, 5, 1, 1, -1, -1, -5, -7, -1, 1, 4, 3, 2, 1, 0, -1, -2, -3, -4, -1, 1, 0, 7, 5, 3, 1, -1, -3, -5, -7, -9, -1

Offset: 0

Stefano Spezia, Dec 02 2023

This array generalizes A367727.

			The array of the fractions begins:
  1,  -1,   -1,   -1,   -1,   -1,    -1,    -1, ...
  1,   0, -1/3, -1/2, -3/5, -2/3,  -5/7,  -3/4, ...
  1, 1/3,    0, -1/5, -1/3, -3/7,  -1/2,  -5/9, ...
  1, 1/2,  1/5,    0, -1/7, -1/4,  -1/3,  -2/5, ...
  1, 3/5,  1/3,  1/7,    0, -1/9,  -1/5, -3/11, ...
  1, 2/3,  3/7,  1/4,  1/9,    0, -1/11,  -1/6, ...
  1, 5/7,  1/2,  1/3,  1/5, 1/11,     0, -1/13, ...
  1, 3/4,  5/9,  2/5, 3/11,  1/6,  1/13,     0, ...
  ...
The array of the numerators begins:
  1, -1, -1, -1, -1, -1, -1, -1, ...
  1,  0, -1, -1, -3, -2, -5, -3, ...
  1,  1,  0, -1, -1, -3, -1, -5, ...
  1,  1,  1,  0, -1, -1, -1, -2, ...
  1,  3,  1,  1,  0, -1, -1, -3, ...
  1,  2,  3,  1,  1,  0, -1, -1, ...
  1,  5,  1,  1,  1,  1,  0, -1, ...
  1,  3,  5,  2,  3,  1,  1,  0, ...
  ...

Stefano Spezia, First 151 antidiagonals of the array

Cf. A367825 (denominator), A367826 (antidiagonal sums).

Cf. A004086, A026741, A051724, A060789, A060819, A106609, A106611, A106612, A106617, A106619, A153881 (n=0), A231190, A367727 (k=1).

A[0,0]=1; A[n_,k_]:=Numerator[(FromDigits[Reverse[IntegerDigits[n]]]-k)/(n+k)]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

A(1, n) = -A026741(n-1) for n > 0.
A(2, n) = -A060819(n-2) for n > 2.
A(3, n) = -A060789(n-3) for n > 3.
A(4, n) = -A106609(n-4) for n > 3.
A(5, n) = -A106611(n-5) for n > 4.
A(6, n) = -A051724(n-6) for n > 5.
A(7, n) = -A106615(n-7) for n > 6.
A(8, n) = -A106617(n-8) = A231190(n) for n > 7.
A(9, n) = -A106619(n-9) for n > 8.
A(10, n) = -A106612(n-10) for n > 9.

A367825 Array read by ascending antidiagonals: A(n, k) is the denominator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 1, 2, 1, 1, 5, 5, 5, 5, 1, 1, 3, 3, 1, 3, 3, 1, 1, 7, 7, 7, 7, 7, 7, 1, 1, 4, 2, 4, 1, 4, 2, 4, 1, 1, 9, 9, 3, 9, 9, 3, 9, 9, 1, 10, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 1, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 4, 6, 12, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1

Offset: 0

Stefano Spezia, Dec 02 2023

This array generalizes A367728.

			The array of the fractions begins:
  1,  -1,   -1,   -1,   -1,   -1,    -1,    -1, ...
  1,   0, -1/3, -1/2, -3/5, -2/3,  -5/7,  -3/4, ...
  1, 1/3,    0, -1/5, -1/3, -3/7,  -1/2,  -5/9, ...
  1, 1/2,  1/5,    0, -1/7, -1/4,  -1/3,  -2/5, ...
  1, 3/5,  1/3,  1/7,    0, -1/9,  -1/5, -3/11, ...
  1, 2/3,  3/7,  1/4,  1/9,    0, -1/11,  -1/6, ...
  1, 5/7,  1/2,  1/3,  1/5, 1/11,     0, -1/13, ...
  1, 3/4,  5/9,  2/5, 3/11,  1/6,  1/13,     0, ...
  ...
The array of the denominators begins:
  1, 1, 1, 1,  1,  1,  1,  1, ...
  1, 1, 3, 2,  5,  3,  7,  4, ...
  1, 3, 1, 5,  3,  7,  2,  9, ...
  1, 2, 5, 1,  7,  4,  3,  5, ...
  1, 5, 3, 7,  1,  9,  5, 11, ...
  1, 3, 7, 4,  9,  1, 11,  6, ...
  1, 7, 2, 3,  5, 11,  1, 13, ...
  1, 4, 9, 5, 11,  6, 13,  1, ...
  ...

Stefano Spezia, First 151 antidiagonals of the array

Cf. A367824 (numerator), A367827 (antidiagonal sums).

Cf. A000012 (n=0), A004086, A026741, A051724, A060789, A060819, A106609, A106611, A106612, A106615, A106617, A231190, A367728 (k=1).

A[0,0]=1; A[n_,k_]:=Denominator[(FromDigits[Reverse[IntegerDigits[n]]]-k)/(n+k)]; Table[A[n-k,k],{n,0,12},{k,0,n}]//Flatten

A(1, n) = A026741(n+1).
A(2, n) = A060819(n+2).
A(3, n) = A060789(n+3).
A(4, n) = A106609(n+4).
A(5, n) = A106611(n+5).
A(6, n) = A051724(n+6).
A(7, n) = A106615(n+7).
A(8, n) = A106617(n+8) = A231190(n+16).
A(9, n) = A106619(n+9).
A(10, n) = A106612(n+10).

A232629 Coefficients of the algebraic number 2sin(4Pi/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

Wolfdieter Lang, Dec 17 2013

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.

			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).

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

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).

cp's OEIS Frontend

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

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A232626 Degree of the algebraic number 2*sin(4*Pi/n).

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A367824 Array read by ascending antidiagonals: A(n, k) is the numerator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

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A367825 Array read by ascending antidiagonals: A(n, k) is the denominator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

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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.

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A232626 Degree of the algebraic number 2sin(4Pi/n).

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