A190184 - OEIS

%I #17 Sep 08 2022 08:45:56

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

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

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

%N Decimal expansion of sqrt(1+x+sqrt(1+2*x)), where x=sqrt(2/3).

%C The rectangle R whose shape (i.e., length/width) is sqrt(1+x+sqrt(1+2x)), where x=sqrt(2/3), can be partitioned into rectangles of shapes sqrt(2) and sqrt(3) in a manner that matches the periodic continued fraction [sqrt(2), sqrt(3), sqrt(2), sqrt(3),...].  R can also be partitioned into squares so as to match the nonperiodic continued fraction [1,1,5,1,6,1,5,1,1,...] at A190185.  For details, see A188635.

%H G. C. Greubel, <a href="/A190184/b190184.txt">Table of n, a(n) for n = 1..10000</a>

%e 1.854493630042558263683640132452778477778...

%t FromContinuedFraction[{2^(1/2), 3^(1/2), {2^(1/2), 3^(1/2)}}]

%t FullSimplify[%]

%t ContinuedFraction[%, 100]  (* A190185 *)

%t RealDigits[N[%%, 120]]      (* A190186 *)

%t N[%%%, 40]

%t RealDigits[Sqrt[1+Sqrt[2/3]+Sqrt[1+2*Sqrt[2/3]]], 10, 100][[1]] (* _G. C. Greubel_, Dec 28 2017 *)

%o (PARI) sqrt(1+sqrt(2/3)+sqrt(1+2*sqrt(2/3))) \\ _G. C. Greubel_, Dec 28 2017

%o (Magma) [Sqrt(1+Sqrt(2/3)+Sqrt(1+2*Sqrt(2/3)))]; // _G. C. Greubel_, Dec 28 2017

%Y Cf. A190185.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, May 05 2011

%E Definition corrected by _Bruno Berselli_, May 13 2011