A189964 - OEIS

%I #13 Sep 08 2022 08:45:56

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

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

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

%N Decimal expansion of (3+x+sqrt(38+6x))/4, where x=sqrt(13).

%C The constant at A189964 is the shape of a rectangle whose continued fraction partition matches [r,r,r,...], where r=(3+sqrt(13))/2. For a general discussion, see A188635. The ordinary continued fraction of r is [3,3,3,3,3,3,3,3,3,3,...]. A rectangle of shape r (that is, (length/width)=r, may be compared with the golden rectangle, with shape [1,1,1,1,1,1,...], and the silver rectangle, with shape [2,2,2,2,2,2,...].

%H G. C. Greubel, <a href="/A189964/b189964.txt">Table of n, a(n) for n = 1..10000</a>

%e 3.5819529507118503770725396959210446869118915483494611612922...

%t r = (3 +13^(1/2))/2;

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

%t FullSimplify[%]

%t N[%, 150]

%t RealDigits[%]  (*A189964*)

%t ContinuedFraction[%%, 120] (*A189965*)

%o (PARI) (3+sqrt(13)+sqrt(38+6*sqrt(13)))/4 \\ _G. C. Greubel_, Jan 12 2018

%o (Magma) (3+Sqrt(13)+Sqrt(38+6*Sqrt(13)))/4 // _G. C. Greubel_, Jan 12 2018

%Y Cf. A188635, A188636, A189965.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, May 04 2011