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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A019918 Decimal expansion of tangent of 20 degrees.

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

Comments

Also the decimal expansion of cotangent of 70 degrees. - Mohammad K. Azarian, Jun 30 2013

Examples

			0.36397023426620236135104788277683404389047...

Links

Crossrefs

Cf. A019938 (tan(2*Pi/9)).

Programs

Formula

Equals tan(Pi/9) = A019829/A019879. - Bernard Schott, Apr 19 2022
Smallest positive of the 6 real roots of x^6-33*x^4+27*x^2-3=0. (Other A019978, A019938). - R. J. Mathar, Aug 31 2025

A353410 a(n) = (tan(1*Pi/9))^(2*n) + (tan(2*Pi/9))^(2*n) + (tan(3*Pi/9))^(2*n) + (tan(4*Pi/9))^(2*n).

Original entry on oeis.org

4, 36, 1044, 33300, 1070244, 34420356, 1107069876, 35607151476, 1145248326468, 36835122753252, 1184744167077204, 38105444942929620, 1225602095970073572, 39419576386043222340, 1267869080483029127412, 40779027899804602385460, 1311593714249667915837060, 42185362424185765127267748
Offset: 0

Views

Author

Bernard Schott, Apr 17 2022

Keywords

Comments

Sum_{k=1..(m-1)/2} (tan(k*Pi/m))^(2*n) is an integer when m >= 3 is an odd integer (see AMM link); this sequence is for the case m = 9.
Note tan(3*Pi/9) = tan(Pi/3) = sqrt(3).

Examples

			a(1) = tan^2 (Pi/9) + tan^2 (2*Pi/9) + tan^2 (3*Pi/9) + tan^2 (4*Pi/9) = 36.

Links

Crossrefs

Similar with: A000244 (m=3), 2*A165225 (m=5), A108716 (m=7), this sequence (m=9), A275546 (m=11), A353411 (m=13).
Cf. A019676 (Pi/9), A019918 (tan(Pi/9)), A019938 (tan(2*Pi/9)).
Cf. A215948.

Programs

  • Mathematica
    LinearRecurrence[{36, -126, 84, -9}, {4, 36, 1044, 33300}, 18] (* Amiram Eldar, Apr 18 2022 *)

Formula

G.f.: 4*(1 - 27x + 63*x^2 - 21*x^3)/((1 - 3*x)*(1 - 33*x + 27*x^2 - 3*x^3)). - Stefano Spezia, Apr 18 2022
a(n) = A215948(n) + 3^n. - Jianing Song, Apr 19 2022

Extensions

More terms from Stefano Spezia, Apr 18 2022
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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