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%I #15 Sep 08 2022 08:45:57
%S 1,4,0,9,5,8,7,9,6,6,7,1,3,2,9,4,7,3,1,5,1,8,2,2,6,4,6,6,1,1,9,6,5,9,
%T 8,7,6,2,4,0,7,3,0,8,8,8,5,9,1,1,5,6,3,5,5,2,8,8,5,5,5,7,2,5,2,1,3,8,
%U 1,6,0,5,3,9,3,2,6,8,3,5,4,3,1,3,3,4,7,9,9,7,9,3,8,8,1,4,6,9,7,6,0,9,9,0,7,0,2,2,6,7,8,6,1,4,5,5,4,4,3,4
%N Decimal expansion of (3 + sqrt(9 + 12x))/6, where x=sqrt(3).
%C The rectangle R whose shape (i.e., length/width) is (3+sqrt(9+12x))/6, where x=sqrt(3), can be partitioned into rectangles of shapes 1 and sqrt(3) in a manner that matches the periodic continued fraction [1, x, 1, x, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [1, 2, 2, 3, 1, 3, 2, 1, 1, 1, ...] at A190263. For details, see A188635.
%H G. C. Greubel, <a href="/A190262/b190262.txt">Table of n, a(n) for n = 1..10000</a>
%e 1.409587966713294731518226466119659876240...
%t r=3^(1/2)
%t FromContinuedFraction[{1, r, {1, r}}]
%t FullSimplify[%]
%t ContinuedFraction[%, 100] (* A190263 *)
%t RealDigits[N[%%, 120]] (* A190262 *)
%t N[%%%, 40]
%t RealDigits[(3 + Sqrt[9 + 12*Sqrt[3]])/6, 10, 100] (* _G. C. Greubel_, Dec 28 2017 *)
%o (PARI) (3 + sqrt(9 + 12*sqrt(3)))/6 \\ _G. C. Greubel_, Dec 28 2017
%o (Magma) [(3 + Sqrt(9 + 12*Sqrt(3)))/6]; // _G. C. Greubel_, Dec 28 2017
%Y Cf. A190263, A188635.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, May 06 2011
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