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A000102
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a(n) = number of compositions of n in which the maximum part size is 4.
(Formerly M1409 N0551)
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3
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0, 0, 0, 0, 1, 2, 5, 12, 27, 59, 127, 269, 563, 1167, 2400, 4903, 9960, 20135, 40534, 81300, 162538, 324020, 644282, 1278152, 2530407, 5000178, 9863763, 19427976, 38211861, 75059535, 147263905, 288609341, 565047233, 1105229439, 2159947998
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OFFSET
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0,6
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COMMENTS
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a(n) is also the number of binary sequences of length n-1 in which the longest run of consecutive 0's is exactly three. - Geoffrey Critzer, Nov 06 2008
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REFERENCES
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J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 155.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. L. Yucas, Counting special sets of binary Lyndon words, Ars Combin., 31 (1991), 21-29.
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LINKS
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FORMULA
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G.f.: x^4/(1 - x - x^2 - x^3)/(1 - x - x^2 - x^3 - x^4).
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EXAMPLE
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For example, a(6)=5 counts 1+1+4, 2+4, 4+2, 4+1+1, 1+4+1. - David Callan, Dec 09 2004
a(6)=5 because there are 5 binary sequences of length 5 in which the longest run of consecutive 0's is exactly 3; 00010, 00011, 01000, 10001, 11000. - Geoffrey Critzer, Nov 06 2008
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MAPLE
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a:= n-> (Matrix(7, (i, j)-> if i+1=j then 1 elif j=1 then [2, 1, 0, -2, -3, -2, -1][i] else 0 fi)^n)[1, 5]: seq(a(n), n=0..40); # Alois P. Heinz, Oct 07 2008
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MATHEMATICA
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a[n_] := MatrixPower[ Table[ Which[i+1 == j, 1, j == 1, {2, 1, 0, -2, -3, -2, -1}[[i]], True, 0], {i, 1, 7}, {j, 1, 7}], n][[1, 5]]; Table[a[n], {n, 0, 34}] (* Jean-François Alcover, May 28 2013, after Alois P. Heinz *)
LinearRecurrence[{2, 1, 0, -2, -3, -2, -1}, {0, 0, 0, 0, 1, 2, 5}, 40] (* Harvey P. Dale, Jul 01 2013 *)
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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