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A190260 Decimal expansion of (1 + sqrt(1 + 2*x))/2, where x=sqrt(2).

%I A190260 #13 Sep 08 2022 08:45:57
%S A190260 1,4,7,8,3,1,8,3,4,3,4,7,8,5,1,5,9,5,6,4,2,2,1,0,4,4,3,6,3,8,5,0,2,2,
%T A190260 2,1,5,2,5,3,2,1,2,1,1,5,0,4,9,9,0,6,4,1,6,7,0,8,4,0,3,9,1,0,2,6,4,9,
%U A190260 9,8,0,5,4,3,7,0,5,7,3,3,2,3,3,6,7,5,1,8,8,2,0,7,4,0,8,2,1,3,6,6,9,7,8,1,0,9,6,7
%N A190260 Decimal expansion of (1 + sqrt(1 + 2*x))/2, where x=sqrt(2).
%C A190260 The rectangle R whose shape (i.e., length/width) is (1+sqrt(1+2x))/2, where x=sqrt(2), can be partitioned into rectangles of shapes 1 and sqrt(2) in a manner that matches the periodic continued fraction [1, x, 1, x, ...].  R can also be partitioned into squares so as to match the nonperiodic continued fraction [1, 2,11,32,1,4,10,2,1,...] at A190261.  For details, see A188635.
%H A190260 G. C. Greubel, <a href="/A190260/b190260.txt">Table of n, a(n) for n = 1..10000</a>
%e A190260 1.478318343478515956422104436385022215253...
%t A190260 r=2^(1/2);
%t A190260 FromContinuedFraction[{1, r, {1, r}}]
%t A190260 FullSimplify[%]
%t A190260 ContinuedFraction[%, 100]  (* A190261 *)
%t A190260 RealDigits[N[%%, 120]]     (* A190260 *)
%t A190260 N[%%%, 40]
%o A190260 (PARI) (1+sqrt(1+2*sqrt(2)))/2 \\ _G. C. Greubel_, Dec 26 2017
%o A190260 (Magma) [(1+Sqrt(1+2*Sqrt(2)))/2]; // _G. C. Greubel_, Dec 26 2017
%Y A190260 Cf. A188635, A190262, A190258.
%K A190260 nonn,cons
%O A190260 1,2
%A A190260 _Clark Kimberling_, May 06 2011