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Decimal expansion of (3+sqrt(109))/10.
3

%I #14 Sep 08 2022 08:45:56

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

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

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

%N Decimal expansion of (3+sqrt(109))/10.

%C Decimal expansion of shape of a (3/5)-extension rectangle; see A188640 for definitions of shape and r-extension rectangle.

%C Briefly, shape=length/width, and an r-extension rectangle is composed of two rectangles of shape 1/r when r<1.

%C The continued fraction of the constant is 1, 2, 1, 9, 1, 2, 1, 1, 2, 1, 9, 1, 2, 1, 1, 2, 1, 9, 1, 2, 1,...

%H G. C. Greubel, <a href="/A188729/b188729.txt">Table of n, a(n) for n = 1..10000</a>

%e 1.3440306508910550179757754022548047678289849837719799753005...

%p evalf((3+sqrt(109))/10,140); # _Muniru A Asiru_, Nov 01 2018

%t r = 3/5; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]

%t N[t, 130]

%t RealDigits[N[t, 130]][[1]]

%t ContinuedFraction[t, 120]

%o (PARI) default(realprecision, 100); (3+sqrt(109))/10 \\ _G. C. Greubel_, Nov 01 2018

%o (Magma) SetDefaultRealField(RealField(100)); (3+Sqrt(109))/10; // _G. C. Greubel_, Nov 01 2018

%Y Cf. A188640, A188658, A188659, A188730.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Apr 10 2011