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.

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A019940 Decimal expansion of tangent of 42 degrees.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Also the decimal expansion of cotangent of 48 degrees. - Ivan Panchenko, Sep 01 2014

Examples

			0.900404044297839945120477203885371702076466211299485282427079...

Links

Crossrefs

Cf. A019851 (sine of 42 degrees)

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); Tan(7*Pi(R)/30); // G. C. Greubel, Nov 25 2018
  • Mathematica
    RealDigits[Tan[42 Degree],10,120][[1]] (* Harvey P. Dale, Sep 05 2012 *)
RealDigits[Tan[7*Pi/30], 10, 100][[1]] (* G. C. Greubel, Nov 25 2018 *)
  • PARI
    default(realprecision, 100); tan(7*Pi/30) \\ G. C. Greubel, Nov 25 2018
  • Sage
    numerical_approx(tan(7*pi/30), digits=100) # G. C. Greubel, Nov 25 2018

Formula

Equals sqrt(7 + 2*sqrt(5) - 2*sqrt(3*(5 + 2*sqrt(5)))). - G. C. Greubel, Nov 25 2018

A019946 Decimal expansion of tangent of 48 degrees.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Also the decimal expansion of cotangent of 42 degrees. - Ivan Panchenko, Sep 01 2014

Examples

			tan(4*Pi/15) = 1.11061251482919287014348196416513553257695951039085904818440222...

Links

Crossrefs

Cf. A019857 (sine of 48 degrees).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); Tan(4*Pi(R)/15); // G. C. Greubel, Nov 24 2018
  • Mathematica
    RealDigits[Tan[48 Degree],10,120][[1]] (* Harvey P. Dale, Nov 26 2011 *)
RealDigits[Tan[4*Pi/15], 10, 100][[1]] (* G. C. Greubel, Nov 24 2018 *)
  • PARI
    default(realprecision, 100); tan(4*Pi/15) \\ G. C. Greubel, Nov 24 2018
  • Sage
    numerical_approx(tan(4*pi/15), digits=100) # G. C. Greubel, Nov 24 2018

Formula

Equals cot(7*Pi/30) = sqrt(23 - 10*sqrt(5) + 2*sqrt(3*(85 -38*sqrt(5)))). - G. C. Greubel, Nov 24 2018
Let r(n) = (n - 1)/(n + 1) if n mod 4 = 1, (n + 1)/(n - 1) otherwise; then this constant equals with Product_{n>=0} r(30*n+15) = (8/7) * (22/23) * (38/37) * (52/53) ... - Dimitris Valianatos, Sep 14 2019
Equals A019857 / A019851. - R. J. Mathar, Sep 06 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

Views

Author

Greg Dresden and Wolfdieter Lang, May 02 2019

Keywords

Comments

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).

Examples

			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
...

Crossrefs

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

Programs

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

Formula

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.
Previous Showing 11-13 of 13 results.

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

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