A337402 Decimal expansion of the length of third shortest diagonal in a regular 12-gon with unit edge length.

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

Offset: 1

Author

Mohammed Yaseen, Aug 26 2020

Keywords

• nonn
• cons

Comments

The distinct diagonal lengths in a regular 12-gon ABC...JKL with unit edge length are

AC = sqrt(2 + sqrt(3)) = sqrt(2)/(-1+sqrt(3)) = A188887

AD = sqrt(4 + 2*sqrt(3)) = 2 /(-1+sqrt(3)) = A090388

AE = sqrt(6 + 3*sqrt(3)) = sqrt(6)/(-1+sqrt(3))

AF = sqrt(7 + 4*sqrt(3)) = (1+sqrt(3))/(-1+sqrt(3)) = A019973

AG = sqrt(8 + 4*sqrt(3)) = 2*sqrt(2)/(-1+sqrt(3)) = A214726

Examples

			3.34606521495123162230117512366749281383748155339375...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane and Gavin A. Theobald, On Dissecting Polygons into Rectangles, arXiv:2309.14866 [math.CO], 2023. See Eq. (2.3).
• I. J. Zucker, G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Cam. Phil. Soc. 131 (2001) 309 eq (8.9)

Crossrefs

Cf. A337301, A188887, A090388, A019973, A214726, A145439.

Programs

Mathematica

First[RealDigits[Sqrt[6+3Sqrt[3]],10,100]] (* Paolo Xausa, Oct 19 2023 *)

PARI

sqrt(6 + 3*sqrt(3)) \\ Michel Marcus, Aug 26 2020

Formula

Equals sin(Pi/3)/sin(Pi/12) = sqrt(2) + 2*cos(Pi/12) = sqrt(3*cot(Pi/12)).

Equals sqrt(6 + 3*sqrt(3)) = sqrt(6)/(-1+sqrt(3)) = (3+sqrt(3))/sqrt(2).

Equals 3*A145439.

Equals Gamma(1/24)*Gamma(11/24)/(Gamma(5/24)*Gamma(7/24)) [Zucker] - R. J. Mathar, Jun 24 2024