A228785 -id:A228785 - cp's OEIS frontend

A267633 Expansion of (1 - 4t)/(1 - x + t x^2): a Fibonacci-type sequence of polynomials.

1, -4, 1, -4, 1, -5, 4, 1, -6, 8, 1, -7, 13, -4, 1, -8, 19, -12, 1, -9, 26, -25, 4, 1, -10, 34, -44, 16, 1, -11, 43, -70, 41, -4, 1, -12, 53, -104, 85, -20, 1, -13, 64, -147, 155, -61, 4, 1, -14, 76, -200, 259, -146, 24

Offset: 0

Tom Copeland, Jan 18 2016

			Row polynomials:
P(0,t) = 1 - 4t
P(1,t) = 1 - 4t = [-t(0) + (1-4t)] = -t(0) + P(0,t)
P(2,t) = 1 - 5t + 4t^2 = [-t(1-4t) + (1-4t)] = -t P(0,t) + P(1,t)
P(3,t) = 1 - 6t + 8t^2 = [-t(1-4t) + (1-5t+4t^2)] = -t P(1,t) + P(2,t)
P(4,t) = 1 - 7t + 13t^2 - 4t^3 = [-t(1-5t+4t^2) + (1-6t+8t^2)]
P(5,t) = 1 - 8t + 19t^2 - 12t^3 = [-t(1-6t+8t^2) + (1-7t+13t^2)]
P(6,t) = 1 - 9t + 26t^2 - 25t^3 + 4t^4
P(7,t) = 1 - 10t + 34t^2 - 44t^3 + 16t^4
P(8,t) = 1 - 11t + 43t^2 - 70t^3 + 41t^4 - 4t^5
P(9,t) = 1 - 12t + 53t^2 - 104t^3 + 85t^4 - 20t^5
P(10,t) = 1 - 13t + 64t^2 - 147t^3 + 155t^4 - 61t^5 + 4t^6
P(11,t) = 1 - 14t + 76t^2 - 200t^3 + 259t^4 - 146t^5 + 24t^6
...
Apparently: The odd rows for n>1 are reversed rows of A140882 (mod signs), and the even rows for n>0, the 9th, 15th, 21st, 27th, etc. rows of A228785 (mod signs). The diagonals are reverse rows of A202241.

Tom Copeland, Addendum to Elliptic Lie Triad

Cf. A000108, A000297, A008574, A008586, A011973, A022088 A034856, A084103, A098474, A124644, A140882, A202241, A228785.

p = (1 - 4 t) / (1 - x + t x^2) + O[x]^12 // CoefficientList[#, x] &;
CoefficientList[#, t] & /@ p // Flatten (* Andrey Zabolotskiy, Mar 07 2024 *)

O.g.f. G(x,t) = (1 - 4t)/(1 - x + t x^2) = a /  [t (x - (1+sqrt(a))/(2t))(x - (1-sqrt(a))/(2t))] with a = 1-4t.
Recursion P(n,t) = -t P(n-2,t) + P(n-1,t) with P(-1,t)=0 and P(0,t) = 1-4t.
Convolution of the Fibonacci polynomials of signed A011973 Fb(n,-t) with coefficients of (1-4t), e.g., (1-4t)Fb(5,-t) = (1-4t)(1-3t+t^2) = 1-7t+13t^2-4t^3, so, for n>=1 and k<=(n-1), T(n,k) = (-1)^k [-4*binomial(n-(k-1),k-1) - binomial(n-k,k)] with the convention that 1/(-m)! = 0 for m>=1, i.e., let binomial(n,k) = nint[n!/((k+c)!(n-k+c)!)] for c sufficiently small in magnitude.
Third column is A034856, and the fourth, A000297. Embedded in the coefficients of the highest order term of the polynomials is A008586 (cf. also A008574).
With P(0,t)=0, the o.g.f. is H(x,t) = (1-4t) x(1-tx)/[1-x(1-tx)] = (1-4t) Linv(Cinv(tx)/t), where Linv(x) = x/(1-x) with inverse L(x) = x/(1+x) and Cinv(x) = x (1-x) is the inverse of the o.g.f. of the shifted Catalan numbers A000108, C(x) = [1-sqrt(1-4x)]/2. Then Hinv(x,t) = C[t L(x/(1-4t))]/t = {1 - sqrt[1-4t(x/(1-4t))/[1+x/(1-4t)]]}/2t = {1-sqrt[1-4tx/(1-4t+x)]}/2t = 1/(1-4t) + (-1+t)/(1-4t)^2 x + (1-2t+2t^2)/(1-4t)^3 x^ + (-1+3t-6t^2+5t^3)/(1-4t)^4 + ..., where the numerators are the signed polynomials of A098474, reverse of A124644.
Row sums (t=1) are periodic -3,-3,0,3,3,0, repeat the six terms ... with o.g.f. -3 - 3x (1-x) / [1-x(1-x)]. Cf. A084103.
Unsigned row sums (t=-1) are shifted A022088 with o.g.f. 5 + 5x(1+x) / [x(1+x)].

Data corrected by Andrey Zabolotskiy, Mar 07 2024

A055035 Degree of minimal polynomial of sin(Pi/n) over the rationals.

Original entry on oeis.org

1, 1, 2, 2, 4, 1, 6, 4, 6, 2, 10, 4, 12, 3, 8, 8, 16, 3, 18, 8, 12, 5, 22, 8, 20, 6, 18, 12, 28, 4, 30, 16, 20, 8, 24, 12, 36, 9, 24, 16, 40, 6, 42, 20, 24, 11, 46, 16, 42, 10, 32, 24, 52, 9, 40, 24, 36, 14, 58, 16, 60, 15, 36, 32, 48, 10, 66, 32, 44, 12, 70, 24, 72

Offset: 1

Shawn Cokus (Cokus(AT)math.washington.edu)

Also degree of minimal polynomial of function F(n)=(gamma(1/n)*gamma((n-1)/n))/Pi over the rationals. Roots of minimal polynomials of F(n) belonging to algebraic extension of sin(n/Pi) and vice versa (e.g. gamma(1/11)*gamma(10/11)/Pi = 20*sin(Pi/11) - 112*sin(Pi/11)^3 + 256*sin(Pi/11)^5 - 256*sin(Pi/11)^7 + (1024*sin(Pi/11)^9)/11). - Artur Jasinski, Oct 17 2011

The algebraic numbers sin(Pi/(2*l)) are given in A228783 in the power basis of the number field Q(2*cos(Pi/(2*l))) if n is even and of Q(2*cos(Pi/l)) if l is odd. In A228785, sin(Pi/(2*l+1)) is given in the power basis of Q(2*cos(Pi/(2*(2*l+1)))) (only odd powers appear). The minimal polynomials for 2*sin(Pi/n), n>=1, are given in A228786. - Wolfdieter Lang, Oct 10 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..175
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

Cf. A000010, A228786 (row length), A093819.

a[n_] := If[n==2, 1, EulerPhi[n]/{1, 1, 2, 1}[[Mod[n, 4]+1]]]; Table[a[n], {n, 80}]
a[n_] := Exponent[ MinimalPolynomial[Sin[Pi/n]][x], x]; Array[a, 75] (* Robert G. Wilson v, Jul 28 2014 *)

a(1)=1, a(2)=1, a(n)=phi(n)/(1, 1, 2, 1 for n=0, 1, 2, 3 mod 4) for n>2, where phi is Euler's totient, A000010

a(n) = A093819(2*n), n >= 1.- Wolfdieter Lang, Oct 29 2019

A228787 Decimal expansion of 2*sin(Pi/17), the ratio side/R in the regular 17-gon inscribed in a circle of radius R.

Original entry on oeis.org

3, 6, 7, 4, 9, 9, 0, 3, 5, 6, 3, 3, 1, 4, 0, 6, 6, 3, 1, 4, 8, 8, 1, 7, 6, 7, 9, 2, 4, 1, 4, 5, 5, 1, 6, 4, 9, 7, 8, 2, 7, 7, 0, 4, 7, 6, 8, 8, 9, 9, 8, 8, 1, 1, 7, 0, 1, 3, 0, 1, 7, 1, 5, 4, 9, 7, 8, 2, 9, 8, 5, 6, 5, 0, 6, 1, 0, 0, 3, 4, 6, 0, 6, 1, 2, 0, 2, 3, 9, 0, 2, 4, 2, 1, 4, 6, 0, 9, 7, 1, 8, 5, 9, 3, 5, 9, 5

Offset: 0

Wolfdieter Lang, Oct 07 2013

s(17) := 2*sin(Pi/17) is an algebraic integer of degree 16 (over the rationals). Its minimal polynomial is 17 - 204*x^2 + 714*x^4 - 1122*x^6 + 935*x^8 - 442*x^10 + 119*x^12 - 17*x^14 + x^16. Its coefficients in the power basis of the algebraic number field Q(2*cos(Pi/34)) are  [0, -15, 0, 140, 0, -378, 0, 450, 0, -275, 0, 90, 0, -15, 0, 1] (see row l = 8 of A228785). The decimal expansion of 2*cos(Pi/34) is given in A228788.
The continued fraction expansion starts with  0; 2, 1, 2, 1, 1, 2, 2, 2, 1, 5, 1, 3, 2, 2, 2, 1, 1, 43, 3, 1, 5, 2, 17, 2, ...
Gauss' formula for cos(2*Pi/17), given in A210644, can be inserted into s(17) = sqrt(2*(1 - cos(2*Pi/17))).
Since 17 is a Fermat prime, this number is constructible and can be written as an expression containing just integers, the basic four arithmetic operations, and square roots. See A003401 for more details. - Stanislav Sykora, May 02 2016

			0.367499035633140663148817679...

Cf. A003401, A019434, A210644, A228785, A228788.

RealDigits[2Sin[Pi/17], 10, 100][[1]] (* Alonso del Arte, Jan 01 2014 *)

2*sin(Pi/17) \\ Charles R Greathouse IV, Nov 12 2014

s(17) = 2*sin(Pi/17) = 2*A241243.

Equals sqrt(34-2*sqrt(17)-2*sqrt(34-2*sqrt(17))-4*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17))))/4. - Stanislav Sykora, May 02 2016

Offset corrected by Rick L. Shepherd, Jan 01 2014

A228786 Table of coefficients of the minimal polynomials of 2*sin(Pi/n), n >= 1.

Original entry on oeis.org

0, 1, -2, 1, -3, 0, 1, -2, 0, 1, 5, 0, -5, 0, 1, -1, 1, -7, 0, 14, 0, -7, 0, 1, 2, 0, -4, 0, 1, -3, 0, 9, 0, -6, 0, 1, -1, 1, 1, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, 1, 0, -4, 0, 1, 13, 0, -91, 0, 182, 0, -156, 0, 65, 0, -13, 0, 1, 1, -2, -1, 1, 1, 0, -8, 0, 14, 0, -7, 0, 1, 2, 0, -16, 0, 20, 0, -8, 0, 1, 17, 0, -204, 0, 714, 0, -1122, 0, 935, 0, -442, 0, 119, 0, -17, 0, 1, 1, -3, 0, 1

Offset: 1

Wolfdieter Lang, Oct 07 2013

s(n) := 2*sin(Pi/n) is, for n >= 2, the length ratio side/R of the regular n-gon inscribed in a circle of radius R. This algebraic number s(n), n >= 1, has the degree gamma(n) := A055035(n), and the row length of this table is gamma(n) + 1.
s(n) has been given in the power basis of the relevant algebraic number field in A228783 for even n (bisected into n == 0 (mod 4) and n == 2 (mod 4)), and in A228785 for odd n.
For the computation of the minimal polynomials ps(n,x), using the coefficients of s(n) in the relevant number field, and the conjugates of the corresponding algebraic numbers rho (giving the length ratios (smallest diagonal)/side in the relevant regular polygons see a comment on A228781. Note that the product of the gamma(n) linear factors (x - conjugates) has to be computed modulo the minimal polynomial of the relevant rho(k) = 2*cos(Pi/k) (called C(k,x=rho(k)) in A187360).
Thanks go to Seppo Mustonen, who asked a question about the square of the sum of all lengths in the regular n-gon, which led to this computation of s(n) and its minimal polynomial.
It would be interesting to find out which length ratios in the regular n-gon give the other positive zeros of the minimal polynomial ps(n,x). See some examples below.
The zeros of the row polynomials ps(n,x) are 2*cos(2*Pi*k/c(2*n))) for gcd(k, c(2*n)) = 1, where c(n) = A178182(n), and k from {0, ..., floor(c(2*n)/2)}, for n >= 1. The number of these solutions is gamma(n) = A055035(n). See the formula section. This results from the zeros of the minimal polynomials of sin(2*Pi/n), with coefficients given in A181872/A181873. - Wolfdieter Lang, Oct 30 2019

			The table a(n, m) starts:
n\m   0  1    2 3   4 5     6 7   8 9   10 12  13 14  15 16 17
1:    0  1
2:   -2  1
3:   -3  0    1
4:   -2  0    1
5:    5  0   -5 0   1
6:   -1  1
7:   -7  0   14 0  -7 0     1
8:    2  0   -4 0   1
9:   -3  0    9 0  -6 0     1
10:  -1  1    1
11: -11  0   55 0 -77 0    44 0 -11 0    1
12:   1  0   -4 0   1
13:  13  0  -91 0 182 0  -156 0  65 0  -13 0   1
14:   1 -2   -1 1
15:   1  0   -8 0  14 0    -7 0   1
16:   2  0  -16 0  20 0    -8 0   1
17:  17  0 -204 0 714 0 -1122 0 935 0 -442  0 119  0 -17  0  1
18:   1 -3    0 1
...
n = 19: [-19, 0, 285, 0, -1254, 0, 2508, 0, -2717, 0, 1729, 0, -665, 0, 152, 0, -19, 0, 1],
n = 20: [1, 0, -12, 0, 19, 0, -8, 0, 1]
n = 5: ps(5,x) = 5 -5*x^2 +1*x^4, with the zeros s(5) = sqrt(3 - tau), sqrt(2 + tau) = tau*s(5) and their negative values, where tau =rho(5) is the golden section. tau*s(5) is the length ratio diagonal/radius in the pentagon.
n = 7: ps(7,x) = -7 + 14*x^2 -7*x^4 + 1*x^6, with the positive zeros s(7) (side/R) about 0.868, s(7)*rho(7) (smallest diagonal/R) about 1.564, and s(7)*(rho(7)^2-1) (longer diagonal/R) about 1.950 in the heptagon inscribed in a circle with radius R.
n = 8: ps(8,x) = 2 -4*x^2 + x^4, with the positive zeros s(8) = sqrt(2-sqrt(2)) and rho(8) = sqrt(2+sqrt(2)) (smallest diagonal/side).
n = 10: ps(10,x) = -1 + x + x^2 with the positive zero s(10) = tau - 1 (the negative solution is -tau).

Cf. A187360, A055035, A228781, A228783, A228785.

a(n, m) = [x^m](minimal polynomial ps(n, x) of 2*sin(Pi/n) over the rationals), n >= 1, m = 0, ..., gamma(n), with gamma(n) = A055035(n).

ps(n,x) = Product_{k=0..floor(c(2*n)/n) and gcd(k, c(2*n)) = 1} (x - 2*cos(2*Pi*k/c(2*n)), with c(2*n) = A178182(2*n), for n >= 1. There are gamma(n) = A055035(n) zeros. - Wolfdieter Lang, Oct 30 2019

A228783 Table of coefficients of the algebraic number s(2l) = 2sin(Pi/2l) as a polynomial in powers of rho(2l) = 2cos(Pi/(2l)) if l is even and of rho(l) = 2*cos(Pi/l) if l is odd (reduced version).

Original entry on oeis.org

2, 0, 1, 1, 0, -3, 0, 1, -1, 1, 0, 4, 0, -1, -1, -1, 1, 0, -7, 0, 14, 0, -7, 0, 1, 2, 1, -1, 0, 8, 0, -18, 0, 8, 0, -1, 1, 2, -3, -1, 1, 0, -8, 0, 6, 0, -1, 0, 0, -1, 3, 3, -4, -1, 1, 0, 12, 0, -67, 0, 96, 0, -52, 0, 12, 0, -1, -2, 3, 1, -1, 0, -15, 0, 140, 0, -378, 0, 450, 0, -275, 0, 90, 0, -15, 0, 1

Offset: 1

Wolfdieter Lang, Oct 06 2013

In the regular (2*l)-gon inscribed in a circle of radius R the length ratio side/R is s(2*l) = 2*sin(Pi/(2*l)). This can be written as a polynomial in the length ratio (smallest diagonal)/side given by rho(2*l) = 2*cos(Pi/(2*l)). (For the 2-gon there is no such diagonal and rho(2) = 0). This leads, in a first step, to the triangle A127672 (see the Oct 05 2013 comment there referring also to the bisections signed A111125 and A127677). Because the minimal polynomial of the algebraic number rho(2*l) of degree delta(2*l) = A055034(2*l), called C(2*l,x) (with coefficients given in A187360) one can eliminate all powers rho(2*l)^k with k >= delta(2*l) by using C(2*l,rho(2*l)) = 0. Furthermore, because for odd l only even powers of rho(2*l) appear, but rho(2*l)^2 = 2 + rho(l), one will obtain a reduced table for s(2*l) with powers rho(2*l)^(2*k+1), k= 0, ..., (delta(2*l)-2)/2 if l is even, and with powers rho(l)^m, m=0, ... , delta(l)-1 if l is odd.
This leads to a reduction of the triangle A127672, when applied for the s(2*l) computation, giving the present table with row length delta(4*L) = A055034(4*L) = phi(8*L)/2 if l =2*L, if L >= 1, and  phi(2*L+1)/2 = A055035(4*L+2), if l = 2*L + 1,  L >= 1, where phi(n) = A000010(n) (Euler totient).
This table gives the coefficients of s(2*l) in the power basis of the algebraic number field Q(rho(2*l)) of degree delta(2*l) = A055034(2*l) if l is even, and in Q(rho(l)) of degree delta(2*l)/2 if l is odd. s(2) and s(6) are rational integers of degree 1.
Thanks go to Seppo Mustonen whose question about the square of the sum of all length in a regular n-gon, led me to this computation.
If l = 2*L+1, L >= 0, that is n == 2 (mod 4), one can obtain s(2*l) more directly in powers of rho(l) by s(2*l) = R(l-1, rho(l)) (mod C(l,rho(l))), with the monic (except for l=1) Chebyshev T-polynomials, called R, in A127672, and the C polynomials from A187360. - Wolfdieter Lang, Oct 10 2013

			The table a(l,m), with n = 2*l, begins:
n,  l \m  0   1   2    3   4   5   6    7   8   9  10  11 ...
2   1:    2
4   2:    0   1
6   3:    1
8   4:    0  -3   0    1
10  5:   -1   1
12  6:    0   4   0   -1
14  7:   -1  -1   1
16  8:    0  -7   0   14   0  -7   0    1
18  9:    2   1  -1
20 10:    0   8   0  -18   0   8   0   -1
22 11:    1   2  -3   -1   1
24 12:    0  -8   0    6   0  -1   0    0
26 13:   -1   3   3   -4  -1   1
28 14:    0  12   0  -67   0  96   0  -52  0  12  0  -1
30 15:   -2   3   1   -1
...
n = 8, l = 4:  s(8)  = -3*rho(8) + rho(8)^3 = -3*sqrt(2 + sqrt(2)) + (sqrt(2 + sqrt(2)))^3 = (sqrt(2) - 1)*sqrt(2 + sqrt(2)).
n = 10, l = 5:  s(10) =  -1 + rho(5), where rho(5) = tau = (1 + sqrt(5))/2, the golden section.

Cf. A127672, A111125, A127677, A055034, A187360, A228785 (odd n case), A228786 (minimal polynomials).

a(2*L,m) = [x^m](s(4*L,x)(mod C(4*L,x))), with s(4*L,x) = sum((-1)^(L-1-s)*A111125(L-1,s)*x^(2*s+1),s=0..L-1), L >= 1, m =0, ..., delta(4*L)-1, and

a(2*L+1,m) = [x^m](s(4*L+2,x)(mod C(2*L+1,x))), with s(4*L+2,x) = sum(A127677(L,s)*(2+x)^(L-s)),s=0..L) (with s(2,x) = 2 for L = 0), L >= 0, m = 0, ..., delta(4*L+2)/2, with delta(n) = A055034(2*l).

A140882 Triangle by rows with row n formed by coefficients of the characteristic polynomial of the n X n tridiagonal matrix with m_{i,i} = 2 for i=1..n, m_{i,i-1} = m_{i,i+1} = -1 for i=2..n-1, and m_{1,2} = m_{n,n-1} = -2.

Original entry on oeis.org

1, 2, -1, 0, -4, 1, 0, -8, 6, -1, 0, -12, 19, -8, 1, 0, -16, 44, -34, 10, -1, 0, -20, 85, -104, 53, -12, 1, 0, -24, 146, -259, 200, -76, 14, -1, 0, -28, 231, -560, 606, -340, 103, -16, 1, 0, -32, 344, -1092, 1572, -1210, 532, -134, 18, -1, 0, -36, 489, -1968, 3630, -3652, 2171, -784, 169, -20, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jul 22 2008

			  1;
  2,  -1;
  0,  -4,   1;
  0,  -8,   6,    -1;
  0, -12,  19,    -8,    1;
  0, -16,  44,   -34,   10,    -1;
  0, -20,  85,  -104,   53,   -12,    1;
  0, -24, 146,  -259,  200,   -76,   14,   -1;
  0, -28, 231,  -560,  606,  -340,  103,  -16,   1;
  0, -32, 344, -1092, 1572, -1210,  532, -134,  18,  -1;
  0, -36, 489, -1968, 3630, -3652, 2171, -784, 169, -20, 1;
  ...

Kemeny, Snell and Thompson, Introduction to Finite Mathematics, 1966, Prentice-Hall, New Jersey, Section 3, Chapter VII, page 407.

Pentti Haukkanen, Jorma Merikoski, and Seppo Mustonen, Some polynomials associated with regular polygons, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.

Cf. A228785, A267633.

Cf. A049310, A133156.

# Assume T(1, 0) = 0 instead of 2.
# Then a slightly modified form of Copeland's second comment gives
# T(n, k) = [t^k] [x^n] gf where
gf := ((t - 4)*t*x^2) / ((t - 2)*x + x^2 + 1) - t*x + 1:
ser := series(gf, x, 12): cx := n -> coeff(ser, x, n):
for n from 0 to 10 do lprint(seq(coeff(cx(n), t, k), k = 0..n)) od;
# Peter Luschny, Apr 27 2024

T[n_, m_, d_] := If[ n == m, 2, If[(n == d && m == d - 1) || ( n == 1 && m == 2), -2, If[(n == m - 1 || n == m + 1), -1, 0]]];
M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}];
a = Join[{{1}}, Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d, 1, 10}]];
Flatten[a]

Starting with the third row (0, -4, 1), these coefficients appear to occur in the odd row polynomials in inverse powers of u for the o.g.f. (u - 4)/(u - u*x + x^2) of A267633, which generates a Fibonacci-type running average of adjacent polynomials of the o.g.f. involving these polynomials and others embedded in A228785. - Tom Copeland, Jan 16 2016

A generating function for the third and later shifted, unsigned rows is (4 + t) / (1 - (2 + t)*x + x^2) = Sum_{n >= 0} (4+t)*U_n((2 + t)/2)*x^n = 4 + t + (8 + 6*t + t^2)*x + ..., where U_n(t) are the Chebyshev polynomials of the second kind of A133156. U_n((2 + t)/2) = V_n((2 + t)), where V_n(t) are the Chebyshev polynomials of A049310. A formula for the coefficients is given by Eqn. 8 in Haukkanen et al. - Tom Copeland, Apr 26 2024

Edited by the editors of the OEIS, Apr 27 2024

cp's OEIS Frontend

A267633 Expansion of (1 - 4t)/(1 - x + t x^2): a Fibonacci-type sequence of polynomials.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A055035 Degree of minimal polynomial of sin(Pi/n) over the rationals.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A228787 Decimal expansion of 2*sin(Pi/17), the ratio side/R in the regular 17-gon inscribed in a circle of radius R.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A228786 Table of coefficients of the minimal polynomials of 2*sin(Pi/n), n >= 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A228783 Table of coefficients of the algebraic number s(2*l) = 2*sin(Pi/2*l) as a polynomial in powers of rho(2*l) = 2*cos(Pi/(2*l)) if l is even and of rho(l) = 2*cos(Pi/l) if l is odd (reduced version).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A140882 Triangle by rows with row n formed by coefficients of the characteristic polynomial of the n X n tridiagonal matrix with m_{i,i} = 2 for i=1..n, m_{i,i-1} = m_{i,i+1} = -1 for i=2..n-1, and m_{1,2} = m_{n,n-1} = -2.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A228783 Table of coefficients of the algebraic number s(2l) = 2sin(Pi/2l) as a polynomial in powers of rho(2l) = 2cos(Pi/(2l)) if l is even and of rho(l) = 2*cos(Pi/l) if l is odd (reduced version).