A188656 - OEIS

%I #19 Apr 28 2021 04:27:33

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

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

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

%N Decimal expansion of (1+sqrt(65))/8.

%C Apart from the second digit the same as A177707.

%C Decimal expansion of the shape of a (1/4)-extension rectangle.

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

%C A (1/4)-extension rectangle matches the continued fraction [1,7,1,1,7,1,1,7,1,1,7,1,1,7,...] for the shape L/W= (1+sqrt(65))/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 (4/3)-extension rectangle, 1 square is removed first, then 7 squares, then 1 square, then 1 square, then 7 squares,..., so that the original rectangle is partitioned into an infinite collection of squares.

%H Daniel Starodubtsev, <a href="/A188656/b188656.txt">Table of n, a(n) for n = 1..10000</a>

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

%e length/width = 1.13278221853731870654582665....

%t RealDigits[(1 + Sqrt[65])/8, 10, 111][[1]] (* _Robert G. Wilson v_, Aug 19 2011 *)

%Y Cf. A188640, A105395.

%K nonn,cons

%O 1,3

%A _Clark Kimberling_, Apr 09 2011