A014097 a(n) = a(n-1)+a(n-4).

1, 1, 1, 5, 6, 7, 8, 13, 19, 26, 34, 47, 66, 92, 126, 173, 239, 331, 457, 630, 869, 1200, 1657, 2287, 3156, 4356, 6013, 8300, 11456, 15812, 21825, 30125, 41581, 57393, 79218, 109343, 150924, 208317, 287535

Offset: 1

Author

David Broadhurst

Keywords

• nonn
• easy

Comments

Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 4 sites wide.

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.

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..7130
• D. J. Broadhurst, Conjectured enumeration of irreducible multiple zeta values, from knots and Feynman diagrams, arXiv:hep-th/9612012, 1996.
• E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1).

Crossrefs

Cf. A020999, A000204, A001609, A000079, A003269, A003520, A005708, A005709, A005710.

Programs

Mathematica

LinearRecurrence[{1,0,0,1},{1,1,1,5},40] (* Harvey P. Dale, Mar 06 2016 *)

Maxima

a(n):=sum(binomial(n-3*j,n-4*j)*n/(n-3*j),j,0,(n-1)/3); /* Vladimir Kruchinin, Mar 25 2016 */

PARI

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 1,0,0,1]^(n-1)*[1;1;1;5])[1,1] \\ Charles R Greathouse IV, Sep 09 2016

Formula

G.f.: -x*(1+4*x^3)/(-1+x+x^4). a(n)= 4*A003269(n)-3*A003269(n-1). - R. J. Mathar, Nov 16 2007

a(n) = Sum_{j=0..(n-1)/3}(binomial(n-3*j,n-4*j)*n/(n-3*j)). - Vladimir Kruchinin, Mar 25 2016

From Greg Dresden, Aug 23 2019: (Start)

a(n) = r1^n + r2^n + r3^n + r4^n, where {r1,r2,r3,r4} are the four roots of x^4-x^3-1=0, see A086106, A230151.

a(n) = round(r^n) for n>21 and r the positive real root of x^4-x^3-1.

(End)

Extensions

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000