A000102 a(n) = number of compositions of n in which the maximum part size is 4. (Formerly M1409 N0551)

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

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