A188887 Decimal expansion of sqrt(2 + sqrt(3)).

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, 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, 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

Offset: 1

Author

Clark Kimberling, Apr 12 2011

Keywords

• nonn
• cons

Comments

Decimal expansion of the length/width ratio of a sqrt(2)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

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.

sqrt(2 + sqrt(3)) is also the shape of the greater sqrt(6)-contraction rectangle; see A188738.

This constant is also the length of the Steiner span of three vertices of a unit square. - Jean-François Alcover, May 22 2014

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

Length of the shortest diagonal in a regular 12-gon with unit side. - Mohammed Yaseen, Nov 12 2020

Examples

			1.931851652578136573499486399457794735267809678016809...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Burkard Polster, Irrational roots, Mathologer video (2018).
• Index entries for algebraic numbers, degree 4.

Crossrefs

Cf. A019884, A019973, A091998, A101263, A188640, A188888, A214726, A217870, A329247.

Programs

Magma

Sqrt(2 + Sqrt(3)); // G. C. Greubel, Apr 10 2018

Mathematica

r = 2^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
RealDigits[Sqrt[2 + Sqrt[3]], 10, 100][[1]] (* G. C. Greubel, Apr 10 2018 *)

PARI

sqrt(2 + sqrt(3)) \\ G. C. Greubel, Apr 10 2018

Formula

Equals (sqrt(6) + sqrt(2))/2.

Equals exp(asinh(cos(Pi/4))). - Geoffrey Caveney, Apr 23 2014

Equals cos(Pi/4) + sqrt(1 + cos(Pi/4)^2). - Geoffrey Caveney, Apr 23 2014

Equals i^(1/6) + i^(-1/6). - Gary W. Adamson, Jul 07 2022

Equals the largest root of x - 1/x = sqrt(2) and of x^2 + 1/x^2 = 4. - Gary W. Adamson, Jun 12 2023

Equals Product_{k>=0} ((12*k + 2)*(12*k + 10))/((12*k + 1)*(12*k + 11)). - Antonio Graciá Llorente, Feb 24 2024

From Amiram Eldar, Nov 23 2024: (Start)

Equals A214726 / 2 = 2 * A019884 = 1 / A101263 = exp(A329247) = A217870^2 = sqrt(A019973).

Equals Product_{k>=1} (1 - (-1)^k/A091998(k)). (End)