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%I #23 Oct 02 2022 22:54:22
%S 3,4,3,2,8,9,2,2,1,5,9,1,3,4,8,3,2,4,4,2,0,1,4,6,0,3,7,0,2,3,5,8,1,0,
%T 9,6,6,9,0,2,7,3,4,1,0,5,8,2,0,2,4,4,4,1,9,5,1,0,1,5,2,2,2,1,9,5,8,7,
%U 9,8,8,1,1,1,4,5,4,4,9,7,0,2,3,0,4,1,2,0,2,4,6,9,6,5,7,3,3,7,8,4,4,6,2,1,6,9,9,3,2,3,2,9,8,3,6,4,2,4,4,3,3,3,0,0,7,2,7,6,8,8
%N Decimal expansion of (Pi+sqrt(4+Pi^2))/2.
%C Decimal expansion of shape of a Pi-extension rectangle; see A188640 for definitions of shape and r-extension rectangle. Briefly, an r-extension rectangle is composed of two rectangles having shape r.
%C A Pi-extension rectangle matches the continued fraction A188723 of the shape L/W = (Pi+sqrt(4+Pi^2))/2. This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for a Pi-extension rectangle, 3 squares are removed first, then 2 squares, then 3 squares, then 4 squares, then 2 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>
%F (Pi+sqrt(4+Pi^2))/2 = [Pi,Pi,Pi,...] (continued fraction). - _Clark Kimberling_, Sep 23 2013
%e 3.4328922159134832442014603702358109669027341058202444195...
%t r = Pi; 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) (Pi+sqrt(4+Pi^2))/2 \\ _Michel Marcus_, Apr 01 2015
%Y Cf. A188640, A188723, A188720, A000796.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Apr 09 2011
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