A246724 Decimal expansion of r_2, the second smallest radius for which a compact packing of the plane exists, with disks of radius 1 and r_2.

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

Offset: 0

Author

Jean-François Alcover, Sep 02 2014

Keywords

• nonn
• cons
• easy

Comments

Essentially the same digit sequence as A176053 and A020832. - R. J. Mathar, Sep 06 2014

This equals the ratio of the radius of the inner Soddy circle and the common radius of the three kissing circles. See A343235, also for links. - Wolfdieter Lang, Apr 19 2021

Previous comment is, together with A176053, the answer to the 1st problem proposed during the 4th Canadian Mathematical Olympiad in 1972. - Bernard Schott, Mar 16 2022

Examples

			0.154700538379251529018297561003914911295203502540253752...
		

References

• Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993 - Canadian Mathematical Society & Société Mathématique du Canada, Problem 1, 1972, page 37, 1993.

Links

• Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2021, p. 62.
• The IMO Compendium, Problem 1, 4th Canadian Mathematical Olympiad, 1972.
• Samuel G. Moreno and Esther M. García, New infinite products of cosines and Viète-like formulae, Mathematics Magazine, Vol. 86, No. 1 (2013), pp. 15-25.
• Bernard Schott, Soddy circles.
• Index entries for algebraic numbers, degree 2.
• Index to sequences related to Olympiads.

Crossrefs

Cf. A246723 (r_1), A246725 (r_3), A246726 (r_4), A246727 (r_5), A002193 (r_6 = sqrt(2)-1), A246728 (r_7), A246729 (r_8), A246730 (r_9).

Cf. A176053, A343235.

Programs

Mathematica

RealDigits[(2*Sqrt[3] - 3)/3, 10, 103] // First

PARI

2/sqrt(3) - 1 \\ Charles R Greathouse IV, Feb 10 2025

Formula

Equals (2*sqrt(3) - 3)/3.

Equals A176053 - 2.

Equals -1 + sqrt(2) * sqrt(2-sqrt(2)) * sqrt(2-sqrt(2-sqrt(2))) * ... (Moreno and García, 2013). - Amiram Eldar, Jun 09 2022