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A216499 The maximum possible number of rooted triples consistent with any galled-tree (level-1 phylogenetic network) containing exactly n leaves.

%I A216499 #21 Jan 19 2019 04:15:43
%S A216499 0,0,0,2,7,16,32,55,87,130,184,252,335,433,550,686,842,1022,1224,1451,
%T A216499 1706,1987,2299,2642,3015,3426,3870,4349,4870,5428,6028,6672,7357,
%U A216499 8091,8871,9696,10576,11505,12486,13525,14616,15766,16976,18242,19574,20968
%N A216499 The maximum possible number of rooted triples consistent with any galled-tree (level-1 phylogenetic network) containing exactly n leaves.
%C A216499 Chao et al. (2012) proved: lim_{n --> infinity} a(n) / (3 (n choose 3)) = 2 (sqrt(3) - 1)/3 = 0.488033... and: a(n) / (3 (n choose 3)) > 2 (sqrt(3) - 1)/3 = 0.488033... for all n > 2.
%C A216499 a(n) = A061061(n) + (n choose 3).
%D A216499 J. Byrka, P. Gawrychowski, K. T. Huber and S. Kelk. Worst-case optimal approximation algorithms for maximizing triple consistency within phylogenetic networks. Journal of Discrete Algorithms, Vol. 8, Number 1, pp. 65-75, 2010.
%D A216499 K.-M. Chao, A.-C. Chu, J. Jansson, R. S. Lemence and A. Mancheron. Asymptotic Limits of a New Type of Maximization Recurrence with an Application to Bioinformatics. Proceedings of the Ninth Annual Conference on Theory and Applications of Models of Computation (TAMC 2012), Lecture Notes in Computer Science, Vol. 7287, pp. 177-188, Springer-Verlag Berlin Heidelberg, 2012.
%D A216499 J. Jansson, N. B. Nguyen and W.-K. Sung. Algorithms for Combining Rooted Triplets into a Galled Phylogenetic Network. SIAM Journal on Computing, Vol. 35, Number 5, pp. 1098-1121, Society for Industrial and Applied Mathematics (SIAM), 2006.
%H A216499 Alois P. Heinz, <a href="/A216499/b216499.txt">Table of n, a(n) for n = 0..1000</a>
%F A216499 a(0) = 0,
%F A216499 a(n) = max_{1<=i<=n} [C(i,3) +2*C(i,2)*(n-i) +i*C(n-i,2) +a(n-i)] for n>0.
%p A216499 a:= proc(n) option remember; `if`(n=0, 0, max(seq(
%p A216499       binomial(i, 3) +2*binomial(i, 2)*(n-i)+
%p A216499       i*binomial(n-i, 2) + a(n-i), i=1..n)))
%p A216499     end:
%p A216499 seq(a(n), n=0..70);  # _Alois P. Heinz_, Jan 28 2016
%t A216499 a[0] = 0; a[n_] := a[n] = Max[Table[Binomial[i, 3] + 2*Binomial[i, 2]*(n-i) + i*Binomial[n-i, 2] + a[n-i], {i, 1, n}]]; Table[a[n], {n, 0, 70}] (* _Jean-François Alcover_, Oct 24 2016 *)
%Y A216499 Cf. A000292 (the analogous sequence for level-0 phylogenetic networks).
%K A216499 nonn
%O A216499 0,4
%A A216499 _Jesper Jansson_, Sep 08 2012