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A079972
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Number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=1, r=4, I={1,2}.
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5
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1, 1, 1, 1, 2, 4, 6, 8, 11, 17, 27, 41, 60, 88, 132, 200, 301, 449, 669, 1001, 1502, 2252, 3370, 5040, 7543, 11297, 16919, 25329, 37912, 56752, 84968, 127216, 190457, 285121, 426841, 639025, 956698, 1432276, 2144238, 3210104, 4805827, 7194801
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OFFSET
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0,5
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COMMENTS
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Number of compositions (ordered partitions) of n into elements of the set {1,4,5}.
a(n+3) is the number of length-n binary words with no substring 1x1 of 1xy1 (that is, no 1's occur with distance two or three), see fxtbook link. - Joerg Arndt, May 27 2011
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REFERENCES
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D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.
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LINKS
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FORMULA
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a(n) = a(n-1) + a(n-4) + a(n-5).
G.f.: 1/(1-x-x^4-x^5).
a(n) = Sum_{k=0..n} Sum_{j=floor((n-k)/4)..floor((n-k)/3)} binomial(k,j)*binomial(j,n-k-3*j). - Vladimir Kruchinin, May 26 2011
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MATHEMATICA
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CoefficientList[Series[1/(1 - x - x^4 - x^5), {x, 0, 41}], x] (* Michael De Vlieger, Feb 03 2020 *)
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PROG
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(Maxima)
a(n):=sum(sum(binomial(k, j)*binomial(j, n-k-3*j), j, floor((n-k)/4), floor((n-k)/3)), k, 0, n); /* Vladimir Kruchinin, May 26 2011 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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