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

1,5

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(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.

Links

Table of n, a(n) for n=1..46.

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).

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).

Example

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

Mathematica

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

Crossrefs

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

Sequence in context: A371565 A353437 A352155 * A321852 A108613 A081408

Adjacent sequences: A058365 A058366 A058367 * A058369 A058370 A058371

Keyword

nonn

Author

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

Status

approved