A027433 Sum over all 2^(2n) pairs (u,v) of binary sequences of length n of length of maximal common subsequence between them.

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

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

Yi Yang, Table of n, a(n) for n = 0..28

Vacláv Chvátal and David Sankoff, Longest Common Subsequences of Two Random Sequences, Journal of Applied Probability, Vol. 12, No. 2 (Jun., 1975), pp. 306-315, 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. Springer-Verlag, 1994, pp. 669-678.

J. D. Dixon, Longest common subsequences in binary sequences, arXiv preprint arXiv:1307.2796 [math.GR], 2013.

Wikipedia, Chvátal-Sankoff constants

Formula

a(n) ~ gamma_2*n*4^n, where gamma_2 is the Chvátal-Sankoff constant.

Mathematica

a[0] = 0;
a[n_] := a[n] = With[{s = Partition[Tuples[{0, 1}, n], 2^(n-1)], 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: A308700 A064837 A224902 * A153338 A007798 A058052

Adjacent sequences: A027430 A027431 A027432 * A027434 A027435 A027436

Keyword

nonn,hard

Author

Paul Zimmermann

Extensions

More terms from Alexander D. Healy, Dec 17 2002

a(19)-a(24) from Yi Yang, Nov 04 2013

a(25)-a(28) from Yi Yang, May 10 2022

Status

approved