This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A188887 #75 Feb 03 2025 12:38:32
%S A188887 1,9,3,1,8,5,1,6,5,2,5,7,8,1,3,6,5,7,3,4,9,9,4,8,6,3,9,9,4,5,7,7,9,4,
%T A188887 7,3,5,2,6,7,8,0,9,6,7,8,0,1,6,8,0,9,1,0,0,8,0,4,6,8,6,1,5,2,6,2,0,8,
%U A188887 4,6,4,2,7,9,5,9,7,1,1,0,3,2,6,9,5,1,2,3,4,8,3,7,1,6,1,4,0,9,0,3,7,7,6,8,0,4,2,2,3,7,2,8,7,6,3,2,4,3,0,7,4,8,9,1,8,5,0,7,5,7
%N A188887 Decimal expansion of sqrt(2 + sqrt(3)).
%C A188887 Decimal expansion of the length/width ratio of a sqrt(2)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.
%C A188887 A sqrt(2)-extension rectangle matches the continued fraction [1,1,13,1,2,15,10,1,18,1,1,21,,...] (A188888) for the shape L/W = sqrt(2 + sqrt(3)). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...]. Specifically, for the sqrt(2)-extension rectangle, 1 square is removed first, then 1 square, then 13 squares, then 1 square, ..., so that the original rectangle of shape sqrt(2 + sqrt(3)) is partitioned into an infinite collection of squares.
%C A188887 sqrt(2 + sqrt(3)) is also the shape of the greater sqrt(6)-contraction rectangle; see A188738.
%C A188887 This constant is also the length of the Steiner span of three vertices of a unit square. - _Jean-François Alcover_, May 22 2014
%C A188887 It is also the larger positive coordinate of (symmetrical) intersection points created by x^2 + y^2 = 4 circle and y = 1/x hyperbola. The smaller coordinate is A101263. - _Leszek Lezniak_, Sep 18 2018
%C A188887 Length of the shortest diagonal in a regular 12-gon with unit side. - _Mohammed Yaseen_, Nov 12 2020
%H A188887 G. C. Greubel, <a href="/A188887/b188887.txt">Table of n, a(n) for n = 1..10000</a>
%H A188887 Burkard Polster, <a href="https://www.youtube.com/watch?v=D6AFxJdJYW4">Irrational roots</a>, Mathologer video (2018).
%H A188887 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.
%F A188887 Equals (sqrt(6) + sqrt(2))/2.
%F A188887 Equals exp(asinh(cos(Pi/4))). - _Geoffrey Caveney_, Apr 23 2014
%F A188887 Equals cos(Pi/4) + sqrt(1 + cos(Pi/4)^2). - _Geoffrey Caveney_, Apr 23 2014
%F A188887 Equals i^(1/6) + i^(-1/6). - _Gary W. Adamson_, Jul 07 2022
%F A188887 Equals the largest root of x - 1/x = sqrt(2) and of x^2 + 1/x^2 = 4. - _Gary W. Adamson_, Jun 12 2023
%F A188887 Equals Product_{k>=0} ((12*k + 2)*(12*k + 10))/((12*k + 1)*(12*k + 11)). - _Antonio Graciá Llorente_, Feb 24 2024
%F A188887 From _Amiram Eldar_, Nov 23 2024: (Start)
%F A188887 Equals A214726 / 2 = 2 * A019884 = 1 / A101263 = exp(A329247) = A217870^2 = sqrt(A019973).
%F A188887 Equals Product_{k>=1} (1 - (-1)^k/A091998(k)). (End)
%e A188887 1.931851652578136573499486399457794735267809678016809...
%t A188887 r = 2^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
%t A188887 N[t, 130]
%t A188887 RealDigits[N[t, 130]][[1]]
%t A188887 ContinuedFraction[t, 120]
%t A188887 RealDigits[Sqrt[2 + Sqrt[3]], 10, 100][[1]] (* _G. C. Greubel_, Apr 10 2018 *)
%o A188887 (PARI) sqrt(2 + sqrt(3)) \\ _G. C. Greubel_, Apr 10 2018
%o A188887 (Magma) Sqrt(2 + Sqrt(3)); // _G. C. Greubel_, Apr 10 2018
%Y A188887 Cf. A019884, A019973, A091998, A101263, A188640, A188888, A214726, A217870, A329247.
%K A188887 nonn,cons
%O A188887 1,2
%A A188887 _Clark Kimberling_, Apr 12 2011