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A190281 Decimal expansion of (1+sqrt(1+r))/r, where r=sqrt(2).

%I A190281 #11 Sep 08 2022 08:45:57
%S A190281 1,8,0,5,7,9,0,8,9,4,6,5,4,3,5,7,4,9,0,4,4,0,6,4,5,5,5,7,3,4,5,5,2,7,
%T A190281 4,1,7,8,2,9,2,2,9,0,5,8,6,1,5,6,3,1,7,8,0,3,3,2,7,5,1,4,4,7,8,3,8,2,
%U A190281 4,1,2,9,2,7,8,6,3,3,8,3,3,0,5,6,1,7,2,9,8,3,3,5,2,0,2,3,6,7,1,1,8,6,6,4,1,2,8,4,3,8,9,2,1,9,0,2,6,9,9,1
%N A190281 Decimal expansion of (1+sqrt(1+r))/r, where r=sqrt(2).
%C A190281 The rectangle R whose shape (i.e., length/width) is (1+sqrt(1+r))/r, where r=sqrt(2), can be partitioned into rectangles of shapes sqrt(2) and 2 in a manner that matches the periodic continued fraction [r, 2, r, 2, ...].  R can also be partitioned into squares so as to match the nonperiodic continued fraction [1,1,4,6,1,2,2,2,1,1 ...] at A190282.  For details, see A188635.
%H A190281 G. C. Greubel, <a href="/A190281/b190281.txt">Table of n, a(n) for n = 1..10000</a>
%e A190281 1.805790894654357490440645557345527417829...
%t A190281 r=2^(1/2)
%t A190281 FromContinuedFraction[{r,2, {r,2}}]
%t A190281 FullSimplify[%]
%t A190281 ContinuedFraction[%, 100]  (* A190282 *)
%t A190281 RealDigits[N[%%, 120]]     (* A190281 *)
%t A190281 N[%%%, 40]
%t A190281 RealDigits[(1 + Sqrt[1 + Sqrt[2]])/Sqrt[2], 10, 100][[1]] (* _G. C. Greubel_, Jan 31 2018 *)
%o A190281 (PARI) (1 + sqrt(1 + sqrt(2)))/sqrt(2) \\ _G. C. Greubel_, Jan 31 2018
%o A190281 (Magma) (1 + Sqrt(1 + Sqrt(2)))/Sqrt(2); // _G. C. Greubel_, Jan 31 2018
%Y A190281 Cf. A190282, A190284.
%K A190281 nonn,cons
%O A190281 1,2
%A A190281 _Clark Kimberling_, May 07 2011