A188657 - OEIS

%I #24 Apr 25 2016 10:52:13

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

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

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

%N Decimal expansion of (3+sqrt(73))/8.

%C Decimal expansion of the length/width ratio of a (3/4)-extension rectangle.

%C See A188640 for definitions of shape and r-extension rectangle for ratio r.

%C A (3/4)-extension rectangle matches the continued fraction [1,2,3,1,7,1,3,2,1,1,2,3,1,7,1,3,2,...] for the shape L/W= (3+sqrt(73))/8. This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...].  Specifically, for the (3/4)-extension rectangle, 1 square is removed first, then 2 squares, then 3 squares, then 1 square, then 7 squares,..., so that the original rectangle is partitioned into an infinite collection of squares.

%H Clark Kimberling, <a href="http://www.jstor.org/stable/27963362">A Visual Euclidean Algorithm</a>, The Mathematics Teacher 76 (1983) 108-109.

%e 1.4430004681646...

%p evalf(3+sqrt(73))/8 ; # _R. J. Mathar_, Apr 11 2011

%t RealDigits[(3 + Sqrt[73])/8, 10, 111][[1]] (* _Robert G. Wilson v_, Aug 19 2011 *)

%o (PARI) (sqrt(73)+3)/8 \\ _Charles R Greathouse IV_, Apr 25 2016

%Y Cf. A188640, A188656.

%K nonn,cons,easy

%O 1,2

%A _Clark Kimberling_, Apr 09 2011