A190287 - OEIS

%I #4 Mar 30 2012 18:57:27

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

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

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

%N Decimal expansion of (5+sqrt(25+4r))/2, where r=sqrt(5).

%C The rectangle R whose shape (i.e., length/width) is (5+sqrt(25+4r))/2, where r=sqrt(5), can be partitioned into rectangles of shapes 5 and r in a manner that matches the periodic continued fraction [5, r, 5, r,  ...].  R can also be partitioned into squares so as to match the nonperiodic continued fraction [5,2,2,2,1,1,1,10,1,1,2,1,...] at A190288.  For details, see A188635.

%e 5.413085645411028710287065567557494135316...

%t r=5^(1/2)

%t FromContinuedFraction[{5, r, {5, r}}]

%t FullSimplify[%]

%t ContinuedFraction[%, 100]  (* A190288 *)

%t RealDigits[N[%%, 120]]     (* A190287 *)

%t N[%%%, 40]

%Y Cf. A188635, A190288.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, May 07 2011