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%I #6 Jun 07 2017 12:27:00
%S 5,4,7,4,1,7,8,4,3,5,8,1,0,7,8,0,6,8,8,1,0,4,9,9,3,2,0,2,0,5,8,6,5,6,
%T 5,8,2,8,4,9,6,0,2,9,3,3,8,3,6,3,4,6,3,2,6,7,2,1,6,9,3,9,3,5,1,8,2,5,
%U 3,3,7,8,0,1,5,4,4,9,7,7,0,5,4,6,1,1,6,7,9,5,5,1,2,9,8,2,6,7,5,6,0,8,5,0,9,2,2,7,0,8,0,0,3,2,2,0,5,7,1,4,5,5,9,3,2,0,2,0,0,0
%N Decimal expansion of sqrt(7)+sqrt(8).
%C Decimal expansion of the length/width ratio of a sqrt(28)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.
%C A sqrt(28)-extension rectangle matches the continued fraction [5,2,9,5,2,687,6,4,1,2,2,...] for the shape L/W=sqrt(7)+sqrt(8). 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(28)-extension rectangle, 5 squares are removed first, then 2 squares, then 9 squares, then 5 squares,..., so that the original rectangle of shape sqrt(7)+sqrt(8) is partitioned into an infinite collection of squares.
%e 5.47417843581078068810499320205865658284960293...
%t r = 28^(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[7]+Sqrt[8],10,150][[1]] (* _Harvey P. Dale_, Jun 07 2017 *)
%Y Cf. A188640, A188933.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Apr 13 2011
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