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 A000102 a(n) = number of compositions of n in which the maximum part size is 4. (Formerly M1409 N0551) 3
 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; text; internal format)
 OFFSET 0,6 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. - Geoffrey Critzer, Nov 06 2008 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. LINKS T. D. Noe, Table of n, a(n) for n=0..200 Nick Hobson, Python program for this sequence J. L. Yucas, Counting special sets of binary Lyndon words, Ars Combin., 31 (1991), 21-29. (Annotated scanned copy) Index entries for linear recurrences with constant coefficients, signature (2, 1, 0, -2, -3, -2, -1). 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). - Franklin T. Adams-Watters, Jan 13 2006 EXAMPLE 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 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); # Alois P. Heinz, Oct 07 2008 MATHEMATICA 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 *) CROSSREFS Sequence in context: A307265 A083378 A116712 * A304175 A086589 A299270 Adjacent sequences: A000099 A000100 A000101 * A000103 A000104 A000105 KEYWORD nonn,nice,easy AUTHOR N. J. A. Sloane EXTENSIONS More terms from Sascha Kurz, Aug 15 2002 Definition improved by David Callan and Franklin T. Adams-Watters STATUS approved

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Last modified November 30 00:17 EST 2023. Contains 367452 sequences. (Running on oeis4.)