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A355118
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The number of nonequivalent root ancestral configurations in a recursively defined family of gene trees and species trees with at least n = 9 leaves, in which for n >= 12 leaves, 3-leaf trees are successively added at the root of the tree with n-3 leaves.
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0
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23, 33, 47, 69, 99, 141, 207, 297, 423, 621, 891, 1269, 1863, 2673, 3807, 5589, 8019, 11421, 16767, 24057, 34263, 50301, 72171, 102789, 150903, 216513, 308367, 452709, 649539, 925101, 1358127, 1948617, 2775303, 4074381, 5845851, 8325909, 12223143, 17537553
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OFFSET
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9,1
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COMMENTS
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a(n) is the number of nonequivalent root ancestral configurations associated with a sequence of pairs (G(n), S(n)), where G(n) is a gene tree and S(n) is a bijectively labeled species tree, G(n) and S(n) both have n leaves, and G(n) = S(n). The sequence of pairs possesses certain (binary, rooted) trees for n = 9, n = 10, and n = 11, as shown in Figure 5 of Disanto and Rosenberg (2019); for n >= 12, the tree G(n) = S(n) is formed by appending the tree G(n-3) and a 3-leaf binary rooted tree to a shared root.
For 9 <= n <= 20, a(n) tabulates the number of nonequivalent root ancestral configurations for the pair (G, S) with the largest number of nonequivalent root ancestral configurations (among pairs with n leaves and G = S). Disanto and Rosenberg (2019) conjecture that a(n) provides this maximum for all n >= 9.
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LINKS
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FORMULA
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a(n) = floor(3^(floor(n/3)) * (2*(n-3*floor(n/3))^2 + 8*(n-3*floor(n/3)) + 23)/27).
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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