A190182 - OEIS

%I #13 Oct 20 2024 02:08:04

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

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

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

%N Decimal expansion of (1+x+sqrt(8+2x))/4, where x=sqrt(15).

%C The rectangle R whose shape (i.e., length/width) is (1+x+sqrt(8+2x))/4, where x=sqrt(15), can be partitioned into golden rectangles and squares in a manner that matches the periodic continued fraction [r,1,1,r,1,1,...]. It can also be partitioned into squares so as to match the nonperiodic continued fraction [2,4,1,3,10,...] at A190183. For details, see A188635.

%H G. C. Greubel, <a href="/A190182/b190182.txt">Table of n, a(n) for n = 1..10000</a>

%e 2.210275532819020968778971352504887053304...

%t r = (1 + 5^(1/2))/2;

%t FromContinuedFraction[{r, 1, 1, {r, 1, 1}}]

%t FullSimplify[%]

%t ContinuedFraction[%, 100] (*A190183*)

%t RealDigits[N[%%, 120]]    (*A190182*)

%t N[%%%, 40]

%t RealDigits[(1 + Sqrt[15] + Sqrt[8 + 2*Sqrt[15]])/4, 10, 100][[1]] (* _G. C. Greubel_, Dec 28 2017 *)

%o (PARI) (1 + sqrt(15) + sqrt(8 + 2*sqrt(15)))/4 \\ _G. C. Greubel_, Dec 28 2017

%o (Magma) [(1 + Sqrt(15) + Sqrt(8 + 2*Sqrt(15)))/4]; // _G. C. Greubel_, Dec 28 2017

%Y Cf. A188635, A190183, A189970.

%K nonn,cons,easy

%O 1,1

%A _Clark Kimberling_, May 05 2011