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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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%I #14 Jun 21 2022 19:22:51

%S 23,33,47,69,99,141,207,297,423,621,891,1269,1863,2673,3807,5589,8019,

%T 11421,16767,24057,34263,50301,72171,102789,150903,216513,308367,

%U 452709,649539,925101,1358127,1948617,2775303,4074381,5845851,8325909,12223143,17537553

%N 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.

%C 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.

%C 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.

%H F. Disanto and N. A. Rosenberg, <a href="https://doi.org/10.1007/s11538-017-0342-x">On the number of non-equivalent ancestral configurations for matching gene trees and species trees</a>, Bull. Math. Biol. 81 (2019), 384-407.

%F a(n) = floor(3^(floor(n/3)) * (2*(n-3*floor(n/3))^2 + 8*(n-3*floor(n/3)) + 23)/27).

%K nonn

%O 9,1

%A _Noah A Rosenberg_, Jun 19 2022