%I
%S 2,0,5,6,9,5,2,4,3,8,7,1,0,9,6,5,9,0,9,3,9,6,7,8,7,9,2,4,3,7,8,8,0,7,
%T 2,5,8,5,8,8,0,9,9,1,4,1,5,4,9,7,1,7,6,2,0,4,6,7,6,4,2,6,8,3,4,1,6,1,
%U 9,5,6,5,7,6,0,3,4,1,7,4,6,1,3,2,2,1,8,2,6,6,1,4,5,7,6,5,0,2,1,5,1,8,9,6,9,9,2,5,3,9,6,2,4,2,1,0,6,6,2,4,8,0,9,8,2,4,8,8,4,2
%N Decimal expansion of shape of a (Pi/2)extension rectangle; shape = (1/4)*(Pi + sqrt(16 + Pi^2).
%C See A188640 for definitions of shape and rextension rectangle. Briefly, an rextension rectangle is composed of two rectangles of shape r.
%C A (Pi/2)extension rectangle matches the continued fraction [2,17,1,1,3,1,3,2,2,1637,1,210,7,...] of the shape L/W = (1/4)*(Pi + sqrt(16 + Pi^2). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for the (Pi/2)extension rectangle, 2 squares are removed first, then 17 squares, then 1 square, then 1 square, then 3 squares, ..., so that the original rectangle is partitioned into an infinite collection of squares.
%e 2.0569524387109659093967879243788072585880991...
%t r = Pi/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]
%Y Cf. A188640, A188722.
%K nonn,easy,cons
%O 1,1
%A _Clark Kimberling_, Apr 09 2011
