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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| 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. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Nov 06 2008]
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REFERENCES
| 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
| T. D. Noe, Table of n, a(n) for n=0..200
Nick Hobson, Python program for this sequence
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FORMULA
| G.f.: x^4/(1-x-x^2-x^3)/(1-x-x^2-x^3-x^4).
a(n)=2*a(n-1)+a(n-2)-2*a(n-4)-3*a(n-5)-2*a(n-6)-a(n-7). Convolution of Tribonacci and Tetranacci numbers (A000073 and A000078). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 13 2006
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EXAMPLE
| For example, a(6)=5 counts 1+1+4, 2+4, 4+2, 4+1+1, 1+4+1. - David Callan (callan(AT)stat.wisc.edu), 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 [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Nov 06 2008]
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MAPLE
| 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); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 07 2008]
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CROSSREFS
| Sequence in context: A129983 A083378 A116712 * A086589 A190171 A091596
Adjacent sequences: A000099 A000100 A000101 * A000103 A000104 A000105
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de), Aug 15 2002
Definition improved by David Callan and Frank Adams-Watters.
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