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%I #1 May 16 2003 03:00:00
%S 1,1,1,1,1,1,1,1,10,11,12,13,14,15,16,17,18,28,39,51,64,78,93,109,126,
%T 144,172,211,262,326,404,497,606,732,876,1048,1259,1521,1847,2251,
%U 2748,3354,4086,4962,6010,7269,8790,10637,12888,15636,18990,23076,28038
%N Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 9 sites wide.
%C This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m-1, a(m) = m+1. The generating function is (x+m*x^m)/(1-x-x^m). Also a(n) = 1 + n*sum(binomial(n-1-(m-1)*i, i-1)/i, i=1..n/m). This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6: A058367, m=7: A058366, m=8: A058365, m=9: A058364.
%D E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
%D Y. Kong, General recurrence theory of ligand binding on a three-dimensional lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799.
%F a(n) = 1 + n*sum(binomial(n-1-8*i, i-1)/i, i=1..n/9). a(n) = a(n-1) + a(n-9), a(n) = 1 for n = 1..8, a(9) = 10. generating function = (x+9*x^9)/(1-x-x^9).
%e a(9) = 10 because there is one way to put zero molecule to the necklace and 9 ways to put one molecule.
%Y Cf. A000079, A003269, A003520, A005708, A005709, A005710.
%K nonn
%O 1,9
%A Yong Kong (ykong(AT)curagen.com), Dec 17 2000
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