OFFSET
1,2
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
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
Burkard Polster, Irrational roots, Mathologer video (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
EXAMPLE
1.9318516525781365734994863994577947352678...
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 *)
PROG
(PARI) sqrt(2 + sqrt(3)) \\ G. C. Greubel, Apr 10 2018
(Magma) Sqrt(2 + Sqrt(3)); // G. C. Greubel, Apr 10 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Apr 12 2011
STATUS
approved