A190157 - OEIS

%I #11 Sep 08 2022 08:45:56

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

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

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

%N Decimal expansion of (1+sqrt(-1+2*sqrt(5)))/2.

%C Let R denote a rectangle whose shape (i.e., length/width) is (1+sqrt(-1+2*sqrt(5)))/2. This rectangle can be partitioned into squares and golden rectangles in a manner that matches the periodic continued fraction [1,r,1,r,1,r,1,r,...], where r is the golden ratio. It can also be partitioned into squares so as to match the nonperiodic continued fraction [1,2,3,6,3,...] at A190158. For details, see A188635.

%H G. C. Greubel, <a href="/A190157/b190157.txt">Table of n, a(n) for n = 1..10000</a>

%e 1.431683416590579253079569133490735199410...

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

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

%t FullSimplify[%]

%t ContinuedFraction[%, 100]  (* A190158 *)

%t RealDigits[N[%%, 120]]     (* A190157 *)

%t N[%%%, 40]

%t RealDigits[(1+Sqrt[-1+2*Sqrt[5]])/2, 10, 100][[1]] (* _G. C. Greubel_, Dec 28 2017 *)

%o (PARI) (1+sqrt(-1+2*sqrt(5)))/2 \\ _G. C. Greubel_, Dec 28 2017

%o (Magma) [(1+Sqrt(-1+2*Sqrt(5)))/2]; // _G. C. Greubel_, Dec 28 2017

%Y Cf. A188635, A190158, A189970, A189971.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, May 05 2011