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
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
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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.