%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 r-extension rectangle. Briefly, an r-extension 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.
%t r = Pi/2; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
%t N[t, 130]
%t RealDigits[N[t, 130]][]
%t ContinuedFraction[t, 120]
%Y Cf. A188640, A188722.
%A _Clark Kimberling_, Apr 09 2011