A248411 Decimal expansion of the best lower bound for 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.

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

Offset: 0

Author

Jean-François Alcover, Oct 06 2014

Keywords

• nonn
• cons

Examples

			x = 0.1486637196311613967236467715222572732594626883945180141...
b = 0.6158277481234066067171143973014413934410965351332132943...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.6 Steiner Tree Constants, p. 504.

Crossrefs

Cf. A220351 (upper bound of rho_3).

Programs

Mathematica

x0 = Root[128*x^6 + 456*x^5 + 783*x^4 + 764*x^3 + 408*x^2 + 108*x - 28, 2]; b = (2 + x0 - Sqrt[x0^2 + x0 + 1])/Sqrt[3]; RealDigits[b, 10, 102] // First

Formula

b = (2 + x - sqrt(x^2 + x + 1))/sqrt(3), where x is the positive root of 128*x^6 + 456*x^5 + 783*x^4 + 764*x^3 + 408*x^2 + 108*x - 28.