A019887 -id:A019887 - cp's OEIS frontend

A306610 a(n) = (2cos(Pi/15))^(-n) + (2cos(7Pi/15))^(-n) + (2cos(11Pi/15))^(-n) + (2cos(13*Pi/15))^(-n), for n >= 1.

4, 24, 109, 524, 2504, 11979, 57299, 274084, 1311049, 6271254, 29997829, 143491199, 686373809, 3283190949, 15704770004, 75121978804, 359337430474, 1718849676159, 8221921677724, 39328626006254, 188124003629279, 899869747188249, 4304424455586134

Offset: 1

Greg Dresden, Feb 28 2019

-a(n) is the coefficient of x in the minimal polynomial for (2*cos(Pi/15))^n, for n >= 1. The coefficients of -x^3 are A306603(n), and those of x^2 are A306611(n).

a(n) is obtained from the Girard-Waring formula for the sum of powers of N = 4 indeterminates (see A324602), with the elementary symmetric functions e_1 = 4, e_2 = -4, e_3 = -1 and e_4 = 1. The arguments are e_j(1/x_1, 1/x_2, 1/x_3, 1/x_4), for j = 1..4, with the zeros {x_i}{i=1..4} of the minimal polynomial of 2*cos(Pi/15), appearing under the negative powers of the formula given above. - _Wolfdieter Lang, May 08 2019

Index entries for linear recurrences with constant coefficients, signature (4,4,-1,-1).

Cf. A019887 (cos(Pi/15)), A019815 (cos(7*Pi/15)), A019851 (cos(11*Pi/15)), A019875 (cos(13*Pi/15)), A306603 (positive powers of these cosines), A306611, A324602.

Table[Round[N[Sum[(2 Cos[k Pi/15])^(-n), {k,{1,7,11,13}}],50]],{n,1,30}]

a(n) = 4a(n-1) + 4a(n-2) - a(n-3) - a(n-4).
G.f.: x*(-4x^3 -3x^2 +8x +4)/(x^4 +x^3 -4x^2 -4x +1).
a(n) = round((2*cos(7*Pi/15))^(-n)) for n >= 3.

A306611 The middle coefficient in the minimal polynomial for (2*cos(Pi/15))^n.

Original entry on oeis.org

-4, 26, -49, 246, -619, 2621, -7774, 30126, -97879, 363131, -1237504, 4497801, -15702574, 56538746, -199764994, 716265246, -2545683874, 9110943101, -32474838004, 116135818131, -414537600379, 1481979727826, -5293483738474, 18921861083121, -67610126265619, 241664630238746

Offset: 1

Greg Dresden, Feb 28 2019

From Wolfdieter Lang, May 01 2019: (Start)
rho(15) = 2*cos(Pi/15) = 2*A019887 gives the length ratio of the smallest diagonal and the side of a regular 15-gon. The minimal polynomial of rho(15) is C(n, x) = x^4 + x^3 - 4*x^2 - 4*x + 1, with zeros x_0 = rho(15), x_1 = 2*cos(7*Pi/15), x_2 = 2*cos(11*Pi/15) and x_3 = 2*cos(13*Pi/15). See A187360, also for a W. Lang link.
The minimal polynomial of rho(1)^n, for n >= 1, considered here, is C(15,n,x) = Product_{j=0..3} (x - x_j^n) = x^4 - A_1(n)x^3 + A_2(n)*x^2 - A_3(n)*x + A_4(n). The coefficients are the elementary symmetric functions A_j(n) = sigma_j((x_0)^n, (x_1)^n, (x_2)^n, (x_3)^n), for j = 1, 2, 3, and A_4(n) = (A_4(1))^n = 1. A_1(n) = A306603(n), A_2(n) = a(n), and A_3(n) = A306610(n), for n >= 1.
Thanks to Greg Dresden for sending me a proof that C(15,n,x) has integer coefficients and does not factor over the rationals for n >= 1. (End)

Index entries for linear recurrences with constant coefficients, signature (-4,5,25,5,-4,-1).

Cf. A306603 (which gives the negative coefficient of x^3 in minimal polynomial for (2 cos(Pi/15))^n) and A306610 (likewise for the coefficient of x).

Cf. A019887 (cos(Pi/15)), A187360.

Table[Coefficient[MinimalPolynomial[(2Cos[Pi/15])^n,x],x,2],{n,1,40}]

Vec(-x*(4 - 10*x - 75*x^2 - 20*x^3 + 20*x^4 + 6*x^5) / ((1 + 3*x + x^2)*(1 + x - 9*x^2 + x^3 + x^4)) + O(x^30)) \\ Colin Barker, Feb 28 2019

a(n) = -4*a(n-1) + 5*a(n-2) + 25*a(n-3) + 5*a(n-4) - 4*a(n-5) - a(n-6).

G.f.: -x*(4 - 10*x - 75*x^2 - 20*x^3 + 20*x^4 + 6*x^5) / ((1 + 3*x + x^2)*(1 + x - 9*x^2 + x^3 + x^4)). - Colin Barker, Feb 28 2019

A343056 Decimal expansion of the real part of i^(1/16), or cos(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

			0.9951847266721968862448369...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Leon D. Fairbanks, Powers of Cosine and Sine, arXiv:2308.04437 [math.GM], 2023. See p. 3.
Wikipedia, Trigonometric constants expressed in real radicals.
Index entries for algebraic numbers, degree 16

cos(Pi/m): A010503 (m=4), A019863 (m=5), A010527 (m=6), A073052 (m=7), A144981 (m=8), A019879 (m=9), A019881 (m=10), A019884 (m=12), A232735 (m=14), A019887 (m=15), A232737 (m=16), A210649 (m=17), A019889 (m=18), A019890 (m=20), A144982 (m=24), A019893 (m=30). this sequence (m=32), A019894 (m=36).

R:= RealField(127); Cos(Pi(R)/32); // G. C. Greubel, Sep 30 2022

RealDigits[Cos[Pi/32], 10, 100][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
real(I^(1/16))
```
PARI
```
cos(Pi/32)
```
PARI
```
sqrt(2+sqrt(2+sqrt(2+sqrt(2))))/2
```

numerical_approx(cos(pi/32), digits=122) # G. C. Greubel, Sep 30 2022

Equals (1/2) * sqrt(2+sqrt(2+sqrt(2+sqrt(2)))).
Satisfies 32768*x^16 -131072*x^14 +212992*x^12 -180224*x^10 +84480*x^8 -21504*x^6 +2688*x^4 -128*x^2 +1 = 0. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/16,1/16;1/2;1/2). - R. J. Mathar, Aug 31 2025

A307886 Array of coefficients of the minimal polynomial of (2*cos(Pi/15))^n, for n >= 1 (ascending powers).

Original entry on oeis.org

1, -4, -4, 1, 1, 1, -24, 26, -9, 1, 1, -109, -49, 1, 1, 1, -524, 246, -29, 1, 1, -2504, -619, -4, 1, 1, -11979, 2621, -99, 1, 1, -57299, -7774, -34, 1, 1, -274084, 30126, -349, 1, 1, -1311049, -97879, -179, 1, 1, -6271254, 363131, -1254, 1, 1, -29997829, -1237504, -824, 1

Offset: 1

Greg Dresden and Wolfdieter Lang, May 02 2019

The length of each row is 5.

The minimal polynomial of (2*cos(Pi/15))^n, for n >= 1, is C(15, n, x) = Product_{j=0..3} (x - (x_j)^n) = Sum_{k=0} T(n, k) x^k, where x_0 = 2*cos(Pi/15), x_1 = 2*cos(7*Pi/15), x_2 = 2*cos(11*Pi/15), and x_3 = 2*cos(13*Pi/15) are the zeros of C(15, 1, x) with coefficients given in A187360 (row n=15).

			The rectangular array T(n, k) begins:
n\k 0      1      2      3      4
---------------------------------
1:  1     -4     -4      1      1
2:  1    -24     26     -9      1
3:  1   -109    -49      1      1
4:  1   -524    246    -29      1
5:  1  -2504   -619     -4      1
6:  1 -11979   2621    -99      1
7:  1 -57299  -7774    -34      1
...

Cf. A019887 (cos(Pi/15)), A019815 (cos(7*Pi/15)), A019851 (cos(11*Pi/15)), A019875 (cos(13*Pi/15)), A187360, A306603, A306610, A306611.

Flatten[Table[CoefficientList[MinimalPolynomial[(2*Cos[\[Pi]/15])^n, x], x], {n, 1, 15}]]

T(n,k) = the coefficient of x^k in C(15, n, x), n >= 1, k=0,1,2,3,4, with C(15, n, k) the minimal polynomial of (2*cos(Pi/15))^n, for n >= 1 as defined above.

T(n, 0) = T(n, 4) = 1. T(n, 1) = -A306610(n), T(n, 2) = A306611(n), T(n, 3) = -A306603(n), n >= 1.

cp's OEIS Frontend

A306610 a(n) = (2*cos(Pi/15))^(-n) + (2*cos(7*Pi/15))^(-n) + (2*cos(11*Pi/15))^(-n) + (2*cos(13*Pi/15))^(-n), for n >= 1.

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A306611 The middle coefficient in the minimal polynomial for (2*cos(Pi/15))^n.

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A343056 Decimal expansion of the real part of i^(1/16), or cos(Pi/32).

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A307886 Array of coefficients of the minimal polynomial of (2*cos(Pi/15))^n, for n >= 1 (ascending powers).

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A306610 a(n) = (2cos(Pi/15))^(-n) + (2cos(7Pi/15))^(-n) + (2cos(11Pi/15))^(-n) + (2cos(13*Pi/15))^(-n), for n >= 1.