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%I #13 Aug 14 2020 13:26:17
%S 1,1,1,1,6,7,8,9,10,16,23,31,40,50,66,89,120,160,210,276,365,485,645,
%T 855,1131,1496,1981,2626,3481,4612,6108,8089,10715,14196,18808,24916,
%U 33005,43720,57916,76724,101640,134645,178365,236281,313005,414645
%N Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 5 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 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 Y. Kong, <a href="http://dx.doi.org/10.1063/1.479242">General recurrence theory of ligand binding on a three-dimensional lattice</a>, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1).
%F a(n) = 1 + n*Sum_{i=1..n/5} binomial(n-4*i-1, i-1)/i.
%F a(n) = a(n-1) + a(n-5) for n >= 6.
%F G.f.: (x+5*x^5)/(1-x-x^5).
%e a(5) = 6 because there is one way to put zero molecule to the necklace and 5 ways to put one molecule.
%t LinearRecurrence[{1,0,0,0,1},{1,1,1,1,6},50] (* _Harvey P. Dale_, Aug 14 2020 *)
%Y Cf. A000204, A001609, A014097, A000079, A003269, A003520, A005708, A005709, A005710.
%K nonn
%O 1,5
%A Yong Kong (ykong(AT)curagen.com), Dec 17 2000
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