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
KEYWORD
nonn,hard
AUTHOR
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