A189967 - OEIS

%I #10 Sep 08 2022 08:45:56

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

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

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

%N Decimal expansion of (7+sqrt(105))/4, which has periodic continued fractions [4,3,4,1,4,3,4,1...] and [7/2, 1, 7/2, 1, ...].

%C Let R denote a rectangle whose shape (i.e., length/width) is (7+sqrt(105))/4.  This rectangle can be partitioned into squares in a manner that matches the continued fraction [4,3,4,1,4,3,4,1...].  It can also be partitioned into rectangles of shape 3/2 and 3 so as to match the continued fraction [7/2, 1, 7/2, 1, ...].  For details, see A188635.

%H G. C. Greubel, <a href="/A189967/b189967.txt">Table of n, a(n) for n = 1..10000</a>

%e 4.311737691489899595805259670130262997684...

%t FromContinuedFraction[{7/2, 1, {7/2, 1}}]

%t ContinuedFraction[%, 25]  (* [4,3,4,1,4,3,4,1...] *)

%t RealDigits[N[%%, 120]]  (* A189968 *)

%t N[%%%, 40]

%o (PARI) (7+sqrt(105))/4 \\ _G. C. Greubel_, Jan 12 2018

%o (Magma) (7+Sqrt(105))/4 // _G. C. Greubel_, Jan 12 2018

%Y Cf. A188635, A189967.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, May 05 2011