A223139 Decimal expansion of (sqrt(13) - 1)/2.

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

Offset: 1

Author

Jaroslav Krizek, Apr 02 2013

Keywords

• nonn
• cons
• easy

Comments

Apart from a(1) the same as A209927 and A085550. [Joerg Arndt, Sep 17 2013]

Decimal expansion of sqrt(3 - sqrt(3 - sqrt(3 - sqrt(3 - ... )))).

Sequence with a(1) = 2 is decimal expansion of sqrt(3 + sqrt(3 + sqrt(3 + sqrt(3 + ... )))) - A209927.

This is the positive root of x^2 + x - 3, and the negative one is -A209927. - Wolfdieter Lang, Aug 29 2022

Examples

			1.3027756377319946465...
		

Links

• B. Sury, Nothing Lucky about 13, Mathematics Magazine, Vol. 83, No. 4 (October 2010), pp. 289-293.

Crossrefs

Cf. A098316, A209927, A085550, A209927.

CF. A122553 (continued fraction).

Programs

Mathematica

RealDigits[(Sqrt[13] - 1)/2, 10, 130]

PARI

(sqrt(13)-1)/2 \\ Altug Alkan, Oct 02 2018

Formula

Closed form: (sqrt(13) - 1)/2 = A209927-1 = A098316-2.

sqrt(3 - sqrt(3 - sqrt(3 - sqrt(3 - ... )))) + 1 = sqrt(3 + sqrt(3 + sqrt(3 + sqrt(3 + ... )))). See A209927.

Equals 2*(cos(2*Pi/13) + cos(6*Pi/13) + cos(8*Pi/13)) (see Sury link). - Michel Marcus, Aug 21 2015