A220351 Decimal expansion of (3*sqrt(3)+sqrt(7))/10.

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

Offset: 0

Author

Charles R Greathouse IV, Dec 11 2012

Keywords

• nonn
• cons

Comments

Smith & Smith conjecture that this is the Steiner ratio rho_3, the least upper bound on the ratio of the length of the Steiner minimal tree to the length of the minimal tree in dimension 3. Diaconis & Graham offer $1000 for proof (or disproof) of this conjecture.

This is an algebraic number of degree 4; the minimal polynomial is 25x^4 - 17x^2 + 1.

Examples

			0.7841903733771222471083954778156877526538674944513592064535755...
		

References

• Persi Diaconis and R. L. Graham, Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks, Princeton University Press, 2011. See pp. 212-214.
• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.6 Steiner Tree Constants, p. 504.

Links

• Warren D. Smith and J. MacGregor Smith, On the Steiner ratio in 3-space, Journal of Combinatorial Theory, Series A 69:2 (1995), pp. 301-332.
• Index entries for algebraic numbers, degree 4

Crossrefs

Cf. A010527.

Programs

Mathematica

RealDigits[(3*Sqrt[3]+Sqrt[7])/10, 10, 120] // First (* Jean-François Alcover, May 27 2014 *)

PARI

(3*sqrt(3)+sqrt(7))/10

PARI

polrootsreal(25*x^4 - 17*x^2 + 1)[4] \\ Charles R Greathouse IV, Jan 05 2016

Formula

(3*sqrt(3)+sqrt(7))/10.

Extensions

Formula and name simplified by Jean-François Alcover, May 27 2014