

A027433


Sum over all 2^(2n) pairs (u,v) of binary sequences of length n of length of maximal common subsequence between them.


1



0, 2, 18, 116, 646, 3324, 16302, 77356, 358424, 1630988, 7317424, 32458400, 142638568, 621948448, 2693978986, 11602817444, 49726594628, 212195409348, 902038055526, 3821542566420, 16141064174876, 67988725603820, 285670814425030, 1197613640781032, 5010423893820844
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OFFSET

0,2


COMMENTS

The proved bounds for gamma_2 (see asymptotic formula below) are 0.788071 <= gamma_2 <= 0.82628, and conjectured value is around 0.811 [see Dixon].


LINKS

Table of n, a(n) for n=0..24.
Vacláv Chvátal and David Sankoff, Longest Common Subsequences of Two Random Sequences, Journal of Applied Probability, Vol. 12, No. 2 (Jun., 1975), pp. 306315, DOI: 10.2307/3212444.
V. Dancik and M. Paterson, Upper bounds for the expected length of a longest common subsequence of two binary sequences, in STACS 94, Proceedings of the Eleventh Annual Symposium on Theoretical Aspects of Computer Science held in Caen, Feb 24 1994. Edited by P. Enjalbert, E. W. Mayr and K. W. Wagner. Lecture Notes in Computer Science, 775. SpringerVerlag, 1994, pp. 669678.
J. D. Dixon, Longest common subsequences in binary sequences, arXiv preprint arXiv:1307.2796 [math.GR], 2013.
Wikipedia, ChvátalSankoff constants


FORMULA

a(n) ~ gamma_2*n*4^n, where gamma_2 is the ChvátalSankoff constant.


MATHEMATICA

a[0] = 0;
a[n_] := a[n] = With[{s = Partition[Tuples[{0, 1}, n], 2^(n1)], f = Composition[Length, LongestCommonSequence]}, 2^n n + 4 Total[ReleaseHold[LowerTriangularize[Outer[Hold[f], s[[1]], s[[1]], 1], 1]], 2] + 2 Total[Outer[f, s[[1]], s[[2]], 1], 2]];
Table[a[n], {n, 0, 10}] (* Vladimir Reshetnikov, May 12 2016 *)


CROSSREFS

Sequence in context: A038721 A064837 A224902 * A153338 A007798 A058052
Adjacent sequences: A027430 A027431 A027432 * A027434 A027435 A027436


KEYWORD

nonn,hard


AUTHOR

Paul Zimmermann


EXTENSIONS

More terms from Alex Healy, Dec 17 2002
a(19)a(24) from Yi Yang, Nov 04 2013


STATUS

approved



