1,5

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.

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).

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).

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

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

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

Sequence in context: A382752 A353437 A352155 * A321852 A108613 A081408

Adjacent sequences: A058365 A058366 A058367 * A058369 A058370 A058371

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

approved