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A014097
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a(n) = a(n-1)+a(n-4).
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10
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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
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OFFSET
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1,4
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COMMENTS
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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(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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FORMULA
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a(n) = Sum_{j=0..(n-1)/3}(binomial(n-3*j,n-4*j)*n/(n-3*j)). - Vladimir Kruchinin, Mar 25 2016
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)
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MATHEMATICA
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LinearRecurrence[{1, 0, 0, 1}, {1, 1, 1, 5}, 40] (* Harvey P. Dale, Mar 06 2016 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
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STATUS
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approved
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