A058368 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 5 sites wide.

1, 1, 1, 1, 6, 7, 8, 9, 10, 16, 23, 31, 40, 50, 66, 89, 120, 160, 210, 276, 365, 485, 645, 855, 1131, 1496, 1981, 2626, 3481, 4612, 6108, 8089, 10715, 14196, 18808, 24916, 33005, 43720, 57916, 76724, 101640, 134645, 178365, 236281, 313005, 414645

Offset: 1

Author

Yong Kong (ykong(AT)curagen.com), Dec 17 2000

Keywords

• nonn

Comments

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_{i=1..n/m} binomial(n-1-(m-1)*i, i-1)/i. 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.

Examples

			a(5) = 6 because there is one way to put zero molecule to the necklace and 5 ways to put one molecule.
		

Links

• E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
• Y. Kong, General recurrence theory of ligand binding on a three-dimensional lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1).

Crossrefs

Cf. A000204, A001609, A014097, A000079, A003269, A003520, A005708, A005709, A005710.

Programs

Mathematica

LinearRecurrence[{1,0,0,0,1},{1,1,1,1,6},50] (* Harvey P. Dale, Aug 14 2020 *)

Formula

a(n) = 1 + n*Sum_{i=1..n/5} binomial(n-4*i-1, i-1)/i.

a(n) = a(n-1) + a(n-5) for n >= 6.

G.f.: (x+5*x^5)/(1-x-x^5).