A188724 - OEIS

%I #19 Jul 17 2021 02:39:14

%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 9,5,6,5,7,6,0,3,4,1,7,4,6,1,3,2,2,1,8,2,6,6,1,4,5,7,6,5,0,2,1,5,1,8,9,6,9,9,2,5,3,9,6,2,4,2,1,0,6,6,2,4,8,0,9,8,2,4,8,8,4,1,9,8

%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.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%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

%E a(130) corrected by _Georg Fischer_, Jul 16 2021