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

0, 0, 0, 2, 7, 16, 32, 55, 87, 130, 184, 252, 335, 433, 550, 686, 842, 1022, 1224, 1451, 1706, 1987, 2299, 2642, 3015, 3426, 3870, 4349, 4870, 5428, 6028, 6672, 7357, 8091, 8871, 9696, 10576, 11505, 12486, 13525, 14616, 15766, 16976, 18242, 19574, 20968

Offset: 0

Author

Jesper Jansson, Sep 08 2012

Keywords

• nonn

Comments

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.

a(n) = A061061(n) + (n choose 3).

References

• 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.
• 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.
• 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.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000292 (the analogous sequence for level-0 phylogenetic networks).

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 0, max(seq(
      binomial(i, 3) +2*binomial(i, 2)*(n-i)+
      i*binomial(n-i, 2) + a(n-i), i=1..n)))
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 28 2016

Mathematica

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 *)

Formula

a(0) = 0,

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.