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A189964 Decimal expansion of (3+x+sqrt(38+6*x))/4, where x=sqrt(13).

%I A189964 #17 Feb 14 2025 03:36:34
%S A189964 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 A189964 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 A189964 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 A189964 Decimal expansion of (3+x+sqrt(38+6*x))/4, where x=sqrt(13).
%C A189964 This constant 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 A189964 G. C. Greubel, <a href="/A189964/b189964.txt">Table of n, a(n) for n = 1..10000</a>
%e A189964 3.5819529507118503770725396959210446869118915483494611612922...
%t A189964 r = (3 +13^(1/2))/2;
%t A189964 FromContinuedFraction[{r, {r}}]
%t A189964 FullSimplify[%]
%t A189964 N[%, 150]
%t A189964 RealDigits[%]  (*A189964*)
%t A189964 ContinuedFraction[%%, 120] (*A189965*)
%o A189964 (PARI) (3+sqrt(13)+sqrt(38+6*sqrt(13)))/4 \\ _G. C. Greubel_, Jan 12 2018
%o A189964 (Magma) (3+Sqrt(13)+Sqrt(38+6*Sqrt(13)))/4 // _G. C. Greubel_, Jan 12 2018
%Y A189964 Cf. A188635, A188636, A189965.
%K A189964 nonn,cons,easy
%O A189964 1,1
%A A189964 _Clark Kimberling_, May 04 2011