|
%I #46 Oct 19 2023 22:40:51
%S 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,
%T 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,
%U 7,8,9,0,6,4,2,1,8,1,8,0,7,1,5,4,5,5,1
%N Decimal expansion of the length of third shortest diagonal in a regular 12-gon with unit edge length.
%C The distinct diagonal lengths in a regular 12-gon ABC...JKL with unit edge length are
%C AC = sqrt(2 + sqrt(3)) = sqrt(2)/(-1+sqrt(3)) = A188887
%C AD = sqrt(4 + 2*sqrt(3)) = 2 /(-1+sqrt(3)) = A090388
%C AE = sqrt(6 + 3*sqrt(3)) = sqrt(6)/(-1+sqrt(3))
%C AF = sqrt(7 + 4*sqrt(3)) = (1+sqrt(3))/(-1+sqrt(3)) = A019973
%C AG = sqrt(8 + 4*sqrt(3)) = 2*sqrt(2)/(-1+sqrt(3)) = A214726
%H Paolo Xausa, <a href="/A337402/b337402.txt">Table of n, a(n) for n = 1..10000</a>
%H N. J. A. Sloane and Gavin A. Theobald, <a href="https://arxiv.org/abs/2309.14866">On Dissecting Polygons into Rectangles</a>, arXiv:2309.14866 [math.CO], 2023. See Eq. (2.3).
%F Equals sin(Pi/3)/sin(Pi/12) = sqrt(2) + 2*cos(Pi/12) = sqrt(3*cot(Pi/12)).
%F Equals sqrt(6 + 3*sqrt(3)) = sqrt(6)/(-1+sqrt(3)) = (3+sqrt(3))/sqrt(2).
%F Equals 3*A145439.
%e 3.34606521495123162230117512366749281383748155339375...
%t First[RealDigits[Sqrt[6+3Sqrt[3]],10,100]] (* _Paolo Xausa_, Oct 19 2023 *)
%o (PARI) sqrt(6 + 3*sqrt(3)) \\ _Michel Marcus_, Aug 26 2020
%Y Cf. A337301, A188887, A090388, A019973, A214726, A145439.
%K nonn,cons
%O 1,1
%A _Mohammed Yaseen_, Aug 26 2020
|
