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%I #12 Sep 08 2022 08:45:56
%S 3,1,7,4,6,7,3,8,9,4,0,3,4,1,9,8,9,2,2,9,5,8,0,7,4,4,1,2,2,1,7,2,4,3,
%T 6,4,2,9,7,4,7,8,6,1,5,8,4,1,2,1,9,6,8,7,2,9,8,3,9,9,1,1,8,5,4,1,0,0,
%U 5,5,6,5,1,4,4,6,7,5,0,7,8,7,0,3,2,2,7,3,6,2,7,3,8,2,3,0,1,0,0,7,3,9,0,6,8,1,8,5,8,2,5,9,5,1,7,6,4,3,9,0
%N Decimal expansion of (1+sqrt(2)+sqrt(7+6*sqrt(2)))/2.
%C Let R denote a rectangle whose shape (i.e., length/width) is (1+sqrt(2)+sqrt(7+6*sqrt(2)))/2. R can be partitioned into squares and silver rectangles in a manner that matches the periodic continued fraction [r,1,r,1,r,1,r,1,...], where r is the silver ratio: 1+sqrt(2)=[2,2,2,2,2,...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [3,5,1,2,1,1,1,2,...] at A190178. For details, see A188635.
%H G. C. Greubel, <a href="/A190177/b190177.txt">Table of n, a(n) for n = 1..10000</a>
%e 3.174673894034198922958074412217243642975...
%t r = 1 + 2^(1/2));
%t FromContinuedFraction[{r, 1, {r, 1}}]
%t FullSimplify[%]
%t ContinuedFraction[%, 100] (* A190178 *)
%t RealDigits[N[%%, 120]] (* A190177 *)
%t N[%%%, 40]
%t RealDigits[(1+Sqrt[2]+Sqrt[7+6*Sqrt[2]])/2, 10, 100][[1]] (* _G. C. Greubel_, Dec 28 2017 *)
%o (PARI) (1+sqrt(2)+sqrt(7+6*sqrt(2)))/2 \\ _G. C. Greubel_, Dec 28 2017
%o (Magma) [(1+Sqrt(2)+Sqrt(7+6*Sqrt(2)))/2]; // _G. C. Greubel_, Dec 28 2017
%Y Cf. A188635, A190178, A189970, A190179.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, May 05 2011
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