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%I #31 Aug 10 2021 06:20:03
%S 1,1,1,5,6,7,8,13,19,26,34,47,66,92,126,173,239,331,457,630,869,1200,
%T 1657,2287,3156,4356,6013,8300,11456,15812,21825,30125,41581,57393,
%U 79218,109343,150924,208317,287535
%N a(n) = a(n-1)+a(n-4).
%C Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 4 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.
%H Indranil Ghosh, <a href="/A014097/b014097.txt">Table of n, a(n) for n = 1..7130</a>
%H D. J. Broadhurst, <a href="http://arXiv.org/abs/hep-th/9612012">Conjectured enumeration of irreducible multiple zeta values, from knots and Feynman diagrams</a>, arXiv:hep-th/9612012, 1996.
%H E. Di Cera and Y. Kong, <a href="http://dx.doi.org/10.1016/S0301-4622(96)02178-3">Theory of multivalent binding in one and two-dimensional lattices</a>, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1).
%F 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
%F a(n) = Sum_{j=0..(n-1)/3}(binomial(n-3*j,n-4*j)*n/(n-3*j)). - _Vladimir Kruchinin_, Mar 25 2016
%F From _Greg Dresden_, Aug 23 2019: (Start)
%F 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.
%F a(n) = round(r^n) for n>21 and r the positive real root of x^4-x^3-1.
%F (End)
%t LinearRecurrence[{1,0,0,1},{1,1,1,5},40] (* _Harvey P. Dale_, Mar 06 2016 *)
%o (Maxima)
%o a(n):=sum(binomial(n-3*j,n-4*j)*n/(n-3*j),j,0,(n-1)/3); /* _Vladimir Kruchinin_, Mar 25 2016 */
%o (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
%Y Cf. A020999, A000204, A001609, A000079, A003269, A003520, A005708, A005709, A005710.
%K nonn,easy
%O 1,4
%A _David Broadhurst_
%E Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
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