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| A216499
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The maximum possible number of rooted triples consistent with any galled-tree (level-1 phylogenetic network) containing exactly n leaves.
(history;
published version)
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#21 by N. J. A. Sloane at Sat Jan 19 04:15:43 EST 2019
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| REFERENCES
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J. Byrka, P. Gawrychowski, K. T. Huber and S. Kelk. Worst-case optimal approximation algorithms for maximizing triplettriple consistency within phylogenetic networks. Journal of Discrete Algorithms, Vol. 8, Number 1, pp. 65-75, 2010.
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Discussion
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| Sat Jan 19
| 04:15
| OEIS Server: https://oeis.org/edit/global/2799
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#20 by N. J. A. Sloane at Sat Jan 19 04:14:58 EST 2019
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| NAME
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The maximum possible number of rooted tripletstriples consistent with any galled-tree (level-1 phylogenetic network) containing exactly n leaves.
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Discussion
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| Sat Jan 19
| 04:14
| OEIS Server: https://oeis.org/edit/global/2798
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#19 by Alois P. Heinz at Mon Oct 24 08:41:08 EDT 2016
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#18 by Jean-François Alcover at Mon Oct 24 08:08:24 EDT 2016
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#17 by Jean-François Alcover at Mon Oct 24 08:08:20 EDT 2016
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| MATHEMATICA
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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 *)
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| STATUS
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approved
editing
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#16 by Alois P. Heinz at Thu Jan 28 18:40:23 EST 2016
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#15 by Alois P. Heinz at Thu Jan 28 18:40:21 EST 2016
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| LINKS
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Alois P. Heinz, <a href="/A216499/b216499.txt">Table of n, a(n) for n = 0..1000</a>
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| MAPLE
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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
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| STATUS
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approved
editing
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#14 by Alois P. Heinz at Thu Jan 28 18:25:52 EST 2016
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#13 by Alois P. Heinz at Thu Jan 28 18:25:45 EST 2016
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| FORMULA
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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.
[(i choose 3) + 2 (i choose 2) (n-i) + i ((n-i) choose 2) + a(n-i)]
for n > 0.
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| STATUS
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approved
editing
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#12 by T. D. Noe at Mon Sep 10 12:09:10 EDT 2012
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