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A073008
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Decimal expansion of the Traveling Salesman constant.
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3
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7, 1, 4, 7, 8, 2, 7, 0, 0, 7, 9, 1, 2, 9, 4, 2, 7, 2, 0, 1, 8, 9, 8, 4, 8, 7, 9, 6, 2, 1, 0, 8, 4, 0, 9, 6, 7, 3, 1, 3, 4, 5, 5, 9, 7, 0, 9, 4, 4, 3, 0, 3, 1, 9, 3, 9, 6, 4, 5, 7, 0, 0, 4, 1, 1, 5, 4, 6, 1, 1, 7, 7, 3, 8, 3, 3, 5, 8, 7, 9, 7, 0, 6, 7, 7, 0, 2, 1, 3, 4, 1, 3, 0, 9, 6, 2, 9, 4, 5, 3, 3, 5, 6, 1, 5
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OFFSET
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0,1
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COMMENTS
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In 1959 J. Beardwood, J. H. Halton and J. M. Hammersley showed that the shortest tour through N random uniformly distributed points in a bounded plane region of area A approaches K*sqrt(N*A), where K is the Traveling Salesman constant, as N approaches infinity. They also proved that 5/8 <= K < 0.922.
In 2015 S. Steinerberger slightly improved both bounds.
In 1995 P. Moscato and N. G. Norman proved that a plane-filling curve called MNPeano is the shortest tour through the set of points defined by MNPeano and observed that the asymptotic expected length of this curve is given by (4/153)*(1+2*sqrt(2))*sqrt(51)*sqrt(N*A), which is very close to the empirical value of the traveling salesman constant.
(End)
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REFERENCES
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J. Beardwood, J. H. Halton and J. M. Hammersley, The shortest path through many points, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 55, No. 4, 1959, pp. 299-327.
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LINKS
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FORMULA
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Conjectured to be equal to (4/153)*(1+2*sqrt(2))*sqrt(51).
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EXAMPLE
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0.7147827007912942720189848796210840967313...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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