A188930 - OEIS

%I #8 Nov 06 2014 04:37:39

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

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

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

%N Decimal expansion of sqrt(5)+sqrt(6).

%C Decimal expansion of the length/width ratio of a sqrt(20)-extension rectangle.  See A188640 for definitions of shape and r-extension rectangle.

%C A sqrt(20)-extension rectangle matches the continued fraction [4,1,2,5,1,1,4,1,2,24,1,2,...] for the shape L/W=sqrt(5)+sqrt(6).  This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...].  Specifically, for the sqrt(20)-extension rectangle, 4 squares are removed first, then 1 square, then 2 squares, then 5 squares,..., so that the original rectangle of shape sqrt(5)+sqrt(6) is partitioned into an infinite collection of squares.

%e 4.6855577202829677946064577434371676274...

%t r = 48^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]

%t N[t, 130]

%t RealDigits[N[t, 130]][[1]]

%t ContinuedFraction[t, 120]

%t RealDigits[Sqrt[5]+Sqrt[6],10,150][[1]] (* _Harvey P. Dale_, Nov 06 2014 *)

%Y Cf. A188640, A188931.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Apr 13 2011