cp's OEIS Frontend

A190258 Decimal expansion of (x + sqrt(2 + 4x))/2, where x=sqrt(2).

%I A190258 #15 Sep 08 2022 08:45:57
%S A190258 2,0,9,0,6,5,7,8,5,0,8,5,2,2,4,4,7,7,5,7,1,0,0,8,9,6,3,5,0,0,5,2,2,1,
%T A190258 3,2,8,0,9,5,8,8,0,1,7,1,5,3,5,0,8,9,6,1,5,2,7,0,1,5,4,0,8,0,1,3,6,5,
%U A190258 3,8,6,8,6,5,8,2,3,0,1,7,6,3,7,1,1,4,3,1,5,0,4,0,4,6,0,4,2,6,3,8,4,6,7,1,8,0,8,3,2,7,8,0,6,7,6,9,3,2,5,8
%N A190258 Decimal expansion of (x + sqrt(2 + 4x))/2, where x=sqrt(2).
%C A190258 The rectangle R whose shape (i.e., length/width) is (x+sqrt(2+4x))/2, where x=sqrt(2), can be partitioned into rectangles of shapes sqrt(2) and 1 in a manner that matches the periodic continued fraction [x, 1, x, 1, ...].  R can also be partitioned into squares so as to match the nonperiodic continued fraction [2,11,32,1,4,10,2,1,...] at A190259.  For details, see A188635.
%H A190258 G. C. Greubel, <a href="/A190258/b190258.txt">Table of n, a(n) for n = 1..10000</a>
%e A190258 2.090657850852244775710089635005221328095...
%t A190258 r=2^(1/2);
%t A190258 FromContinuedFraction[{r,1, {r,1}}]
%t A190258 FullSimplify[%]
%t A190258 ContinuedFraction[%, 100]  (* A190258 *)
%t A190258 RealDigits[N[%%, 120]]     (* A190259 *)
%t A190258 N[%%%, 40]
%t A190258 RealDigits[(Sqrt[2]+Sqrt[2+4Sqrt[2]])/2,10,120][[1]] (* _Harvey P. Dale_, Jun 20 2021 *)
%o A190258 (PARI) sqrt(1/2)+sqrt(1/2+sqrt(2))
%o A190258 (Magma) [(Sqrt(2) + Sqrt(2+4*Sqrt(2)))/2]; // _G. C. Greubel_, Dec 26 2017
%Y A190258 Cf. A190259, A190260, A188635.
%K A190258 nonn,easy,cons
%O A190258 1,1
%A A190258 _Clark Kimberling_, May 06 2011