A174968 Decimal expansion of (1 + sqrt(2))/2.

1, 2, 0, 7, 1, 0, 6, 7, 8, 1, 1, 8, 6, 5, 4, 7, 5, 2, 4, 4, 0, 0, 8, 4, 4, 3, 6, 2, 1, 0, 4, 8, 4, 9, 0, 3, 9, 2, 8, 4, 8, 3, 5, 9, 3, 7, 6, 8, 8, 4, 7, 4, 0, 3, 6, 5, 8, 8, 3, 3, 9, 8, 6, 8, 9, 9, 5, 3, 6, 6, 2, 3, 9, 2, 3, 1, 0, 5, 3, 5, 1, 9, 4, 2, 5, 1, 9, 3, 7, 6, 7, 1, 6, 3, 8, 2, 0, 7, 8, 6, 3, 6, 7, 5, 0

Offset: 1

Author

Klaus Brockhaus, Apr 02 2010

Keywords

• cons
• nonn
• easy
• changed

Comments

a(n) is the diameter of the circle around the Vitruvian Man when the square has sides of unit length. See illustration in links. - Kival Ngaokrajang, Jan 29 2015

The iterated function z^2 - 1/4, starting from z = 0, gives a pretty good rational approximation of (-1)((1 + sqrt(2))/2 - 1) to more than eight decimal digits after just twenty steps. - Alonso del Arte, Apr 09 2016

This sequence describes the minimum Euclidean length of the optimal solution of the well-known Nine dots puzzle, published in Sam Loyd’s Cyclopedia of puzzles (1914), p. 301 since a valid polygonal chain satisfying the conditions of the above-mentioned problem is (0, 1)-(0, 3)-(3, 0)-(0, 0)-(2, 2), and its total length is equal to 5*(1 + sqrt(2)) = 12.071... (i.e., 10*(1 + sqrt(2))/2). - Marco Ripà, Jul 22 2024

Examples

			1.20710678118654752440084436210484903928483593768847...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Kival Ngaokrajang, Illustration of Vitruvian Man and construction rule.
• Marco Ripà, Shortest polygonal chains covering each planar square grid, arXiv:2207.08708v3 [math.CO], 2024.
• Wikipedia, Vitruvian Man.
• Index entries for algebraic numbers, degree 2.

Crossrefs

Cf. A002193 (decimal expansion of sqrt(2)), A010685 (continued fraction expansion of (1 + sqrt(2))/2), A079291, A249403.

Apart from initial digits the same as A157214 and A010503.

Programs

Mathematica

First@RealDigits[(1 + Sqrt[2])/2, 10, 105] (* Michael De Vlieger, Jan 29 2015 *)

PARI

(1+sqrt(2))/2 \\ Altug Alkan, Apr 16 2016

Formula

Equals Product_{k>=2} (1 + (-1)^k/A079291(k)). - Amiram Eldar, Dec 03 2024