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%I #23 Feb 27 2023 11:30:51
%S 2,1,8,8,9,0,1,0,5,9,3,1,6,7,3,3,9,4,2,0,1,4,5,3,1,0,4,7,5,7,2,5,6,6,
%T 3,9,6,3,2,6,5,3,2,2,1,8,4,4,6,4,1,5,4,0,4,2,1,2,0,7,0,7,1,9,3,2,6,5,
%U 0,0,9,2,0,0,6,9,5,4,1,8,3,2,4,2,0,7,6,9,5,3,6,6,1,5,8,9,6,0,9,3,1,4,5,3,4,5,3,5,9,8,7,6,9,5,2,0,8,3,0,6,2,8,5,6,7,3,7,4,9
%N Decimal expansion of (sqrt(3) + sqrt(7))/2.
%C Decimal expansion of the length/width ratio of a sqrt(3)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.
%C A sqrt(3)-extension rectangle matches the continued fraction [2,5,3,2,2,9,1,2,1,2,1,9,...] for the shape L/W=(sqrt(3)+sqrt(7))/2. This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...]. Specifically, for the sqrt(3)-extension rectangle, 2 squares are removed first, then 5 squares, then 3 squares, then 2 squares, ..., so that the original rectangle of shape (sqrt(3)+sqrt(7))/2 is partitioned into an infinite collection of squares.
%F (sqrt(3)+sqrt(7))/2 = exp(asinh(cos(Pi/6))). - _Geoffrey Caveney_, Apr 23 2014
%F cos(Pi/6) + sqrt(1+cos(Pi/6)^2). - _Geoffrey Caveney_, Apr 23 2014
%e 2.1889010593167339420145310475725663963265322184...
%t r = 3^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
%t N[t, 130]
%t RealDigits[N[t, 130]][[1]]
%t ContinuedFraction[t, 120]
%t RealDigits[(Sqrt[3]+Sqrt[7])/2,10,140][[1]] (* _Harvey P. Dale_, Feb 27 2023 *)
%Y Cf. A188640, A188923.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Apr 13 2011
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