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A350361
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2-tone chromatic number of a tree with maximum degree n.
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5
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4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15
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OFFSET
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1,1
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COMMENTS
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The 2-tone chromatic number of a graph G is the smallest number of colors for which G has a coloring where every vertex has two distinct colors, no adjacent vertices have a common color, and no pair of vertices at distance 2 have two common colors.
a(n) is also the 2-tone chromatic number of a star with n leaves.
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LINKS
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FORMULA
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a(n) = ceiling((5 + sqrt(1 + 8*n))/2).
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EXAMPLE
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For a star with three leaves, label the leaves 12, 13, and 23. Label the other vertex 45. A total of 5 colors are used, so a(3)=5.
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MATHEMATICA
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Table[Ceiling[(5 + Sqrt[1 + 8*n])/2], {n, 71}] (* Stefano Spezia, Dec 27 2021 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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