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A058368
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Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 5 sites wide.
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9
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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
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OFFSET
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1,5
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COMMENTS
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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.
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LINKS
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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).
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FORMULA
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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).
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EXAMPLE
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a(5) = 6 because there is one way to put zero molecule to the necklace and 5 ways to put one molecule.
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MATHEMATICA
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LinearRecurrence[{1, 0, 0, 0, 1}, {1, 1, 1, 1, 6}, 50] (* Harvey P. Dale, Aug 14 2020 *)
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CROSSREFS
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Cf. A000204, A001609, A014097, A000079, A003269, A003520, A005708, A005709, A005710.
Sequence in context: A115840 A353437 A352155 * A321852 A108613 A081408
Adjacent sequences: A058365 A058366 A058367 * A058369 A058370 A058371
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KEYWORD
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nonn
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AUTHOR
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Yong Kong (ykong(AT)curagen.com), Dec 17 2000
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STATUS
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approved
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