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%I #6 Oct 19 2021 18:56:20
%S 3,4,9,5,5,0,7,6,5,6,6,0,4,9,2,4,5,0,3,7,7,2,8,6,6,6,7,9,0,5,4,4,8,1,
%T 0,0,5,1,8,8,6,1,0,8,8,4,0,4,7,7,3,2,4,5,3,1,4,2,1,1,4,5,0,6,9,7,2,1,
%U 5,2,8,1,4,6,2,1,9,5,7,6,9,3,6,8,9,3,0,5,8,5,3,9,9,3,9,4,4,9,0,9,1,2,9,6,6,7,2,8,9,9,6,0,0,8,6,3,6,9,8,1
%N Decimal expansion of (3+sqrt(9+4r))/2, where r=sqrt(3).
%C The rectangle R whose shape (i.e., length/width) is (3+sqrt(9+4r))/2, where r=sqrt(3), can be partitioned into rectangles of shapes 3 and r in a manner that matches the periodic continued fraction [3, r, 3, r, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [3,2,55,6,1,1,1,9,1,1,1,7,2,...] at A190286. For details, see A188635.
%e 3.495507656604924503772866679054481005189...
%t r=3^(1/2)
%t FromContinuedFraction[{3, r, {3, r}}]
%t FullSimplify[%]
%t ContinuedFraction[%, 100] (* A190286 *)
%t RealDigits[N[%%, 120]] (* A190285 *)
%t N[%%%, 40]
%t RealDigits[(3+Sqrt[9+4Sqrt[3]])/2,10,120][[1]] (* _Harvey P. Dale_, Oct 19 2021 *)
%Y Cf. A188635, A190286.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, May 07 2011
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