A189968 - OEIS

%I #14 Sep 08 2022 08:45:56

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

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

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

%N Decimal expansion of (5+sqrt(85))/6, which has periodic continued fractions [2,2,1,2,2,1,...] and [5/2, 1, 5/2, 1, ...].

%C Let R denote a rectangle whose shape (i.e., length/width) is (5+sqrt(85))/6.  This rectangle can be partitioned into squares in a manner that matches the continued fraction [2,2,1,2,2,1,...].  It can also be partitioned into rectangles of shape 3/2 and 3 so as to match the continued fraction [5/2, 1, 5/2, 1, ...].  For details, see A188635.

%H G. C. Greubel, <a href="/A189968/b189968.txt">Table of n, a(n) for n = 1..10000</a>

%e 2.369924076215481218333712380293798859541...

%t FromContinuedFraction[{5/2, 1, {5/2, 1}}]

%t ContinuedFraction[%, 25]  (* [2,2,1,2,2,1,...] *)

%t RealDigits[N[%%, 120]]  (* A189967 *)

%t N[%%%, 40]

%t RealDigits[(5+Sqrt[85])/6,10,120][[1]] (* _Harvey P. Dale_, Apr 18 2014 *)

%o (PARI) (5+sqrt(85))/6 \\ _G. C. Greubel_, Jan 12 2018

%o (Magma) (5+Sqrt(85))/6 // _G. C. Greubel_, Jan 12 2018

%Y Cf. A188635, A189966.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, May 05 2011