A006979 a(n) is the number of compositions of n in which the maximum part size is 5. (Formerly M1410)

0, 0, 0, 0, 0, 1, 2, 5, 12, 28, 63, 139, 303, 653, 1394, 2953, 6215, 13008, 27095, 56201, 116143, 239231, 491326, 1006420, 2056633, 4193706, 8534653, 17337764, 35162804, 71205504, 143990366, 290795624, 586566102, 1181834852, 2378701408

Offset

0,7

Comments

a(n) is also the number of binary sequences of length n-1 in which the longest run of 0's is exactly 4. Example: a(7) = 5 because there are 5 binary sequences of length 6 in which the longest run of 0's is exactly 4: 000010, 000011, 010000, 110000, 100001. - Geoffrey Critzer, Nov 07 2008

References

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

Matthew House, Table of n, a(n) for n = 0..3390

Index entries for linear recurrences with constant coefficients, signature (2,1,0,-1,-3,-4,-3,-2,-1).

Formula

G.f.: x^5 / ((1-x-x^2-x^3-x^4)*(1-x-x^2-x^3-x^4-x^5)). - Alois P. Heinz, Oct 29 2008

Maple

a:= n-> (Matrix(9, (i, j)-> if i=j-1 then 1 elif j=1 then [2, 1, 0, -1, -3, -4, -3, -2, -1][i] else 0 fi)^n) [1, 6]: seq(a(n), n=0..40); # Alois P. Heinz, Oct 29 2008

Mathematica

CoefficientList[Series[x^5/((1 - x - x^2 - x^3 - x^4) (1 - x - x^2 - x^3 - x^4 - x^5)), {x, 0, 34}], x] (* Michael De Vlieger, Feb 11 2017 *)

Crossrefs

Cf. A048003.

Sequence in context: A111586 A192657 A320590 * A019301 A006980 A045623

Adjacent sequences: A006976 A006977 A006978 * A006980 A006981 A006982

Keyword

nonn

Author

Simon Plouffe

Extensions

More terms and better definition from Alois P. Heinz, Oct 29 2008

Offset corrected by Matthew House, Feb 11 2017

Status

approved