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%I #11 Sep 08 2022 08:45:56
%S 2,7,0,3,2,5,7,4,0,9,5,4,8,8,1,4,5,5,1,6,6,7,0,4,5,7,1,3,6,2,7,1,3,2,
%T 1,9,2,8,7,4,4,6,7,5,0,8,1,2,0,4,1,0,6,6,8,0,0,1,2,9,2,0,3,4,2,4,0,4,
%U 4,5,1,7,1,1,3,3,6,4,5,9,1,0,1,2,7,9,8,2,3,4,8,4,6,5,5,4,6,7,6,0,8,2,3,3,8,9,9,6,8,1,4,6,4,7,8,6,1,4,0,2,5,3,5,4,1,1,0,5,5,7
%N Decimal expansion of (7+sqrt(85))/6.
%C Decimal expansion of the length/width ratio of a (7/3)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.
%C A (7/3)-extension rectangle matches the continued fraction [2,1,2,2,1,2,2,1,2,2,1,...] for the shape L/W=(7+sqrt(85))/6. 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 (7/3)-extension rectangle, 2 squares are removed first, then 1 square, then 2 squares, then 2 squares,..., so that the original rectangle of shape (7+sqrt(85))/6 is partitioned into an infinite collection of squares.
%H G. C. Greubel, <a href="/A188737/b188737.txt">Table of n, a(n) for n = 1..10000</a>
%e 2.703257409548814551667045713627132192874467508120...
%p evalf((7+sqrt(85))/6,140); # _Muniru A Asiru_, Nov 01 2018
%t r = 7/3; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
%t N[t, 130]
%t RealDigits[N[t, 130]][[1]]
%t ContinuedFraction[t, 120]
%o (PARI) default(realprecision, 100); (7+sqrt(85))/6 \\ _G. C. Greubel_, Nov 01 2018
%o (Magma) SetDefaultRealField(RealField(100)); (7+Sqrt(85))/6; // _G. C. Greubel_, Nov 01 2018
%Y Cf. A188640.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Apr 12 2011
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