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

1,1

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

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Amrik Singh Nimbran, Interesting infinite products of rational functions motivated by Euler, Math. Student, Vol. 85 (2016), pp. 117-133; ResearchGate link.

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 and G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Cam. Phil. Soc. 131 (2001), 309-319, p. 309, eq. (8.9).

Index entries for algebraic numbers, degree 4.

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

Minimal polynomial: x^4 - 12*x^2 + 9. - Amiram Eldar, Jun 06 2026

Equals Product_{k>=1} (12*k-8)*(12*k-4)/((12*k-11)*(12*k-1)) = Product_{k>=1} (24*k-19)*(24*k-17)/((24*k-23)*(24*k-13)) (Nimbran, 2016, p. 131, eq. (58)). - Amiram Eldar, Jun 28 2026

Example

3.34606521495123162230117512366749281383748155339375...

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A373721 A043551 A162888 * A151759 A008443 A196456

Adjacent sequences: A337399 A337400 A337401 * A337403 A337404 A337405

Keyword

nonn,cons,changed

Author

Mohammed Yaseen, Aug 26 2020

Status

approved