A006980 Compositions: 6th column of A048004. (Formerly M1411)

1, 2, 5, 12, 28, 64, 143, 315, 687, 1485, 3186, 6792, 14401, 30391, 63872, 133751, 279177, 581040, 1206151, 2497895, 5161982, 10646564, 21919161, 45052841, 92461171, 189489255, 387830160, 792810956, 1618840800, 3301999647

Offset

6,2

Comments

a(n) is the number of binary strings of length n-1 whose longest run of 1s has length 5. - Félix Balado, May 20 2025

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

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Félix Balado and Guénolé C. M. Silvestre, Systematic Enumeration of Fundamental Quantities Involving Runs in Binary Strings, arXiv:2602.10005 [math.CO], 2026. See p. 30.

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,-1,-2,-4,-5,-4,-3,-2,-1).

Formula

G.f.: x^6 / ((1-x-x^2-x^3-x^4-x^5) * (1-x-x^2-x^3-x^4-x^5-x^6)). - Alois P. Heinz, Oct 29 2008

G.f.: x^6 * (1-x)^2 / ((1-2*x+x^6) * (1-2*x+x^7)). - Félix Balado, May 20 2025

Maple

a:= n-> (Matrix(11, (i, j)-> if i=j-1 then 1 elif j=1 then [2, 1, 0, -1, -2, -4, -5, -4, -3, -2, -1][i] else 0 fi)^n) [1, 7]: seq(a(n), n=6..40); # Alois P. Heinz, Oct 29 2008

Prog

(PARI) Vec(1/(1-x-x^2-x^3-x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6)+O(x^99)) \\ Charles R Greathouse IV, Jan 10 2013

Crossrefs

Cf. A048004, A006979, A000102, A000100, A000071.

Sequence in context: A320590 A006979 A019301 * A045623 A290990 A324586

Adjacent sequences: A006977 A006978 A006979 * A006981 A006982 A006983

Keyword

nonn,easy

Author

Simon Plouffe

Extensions

Corrected definition: 6th column of A048004. - Geoffrey Critzer, Nov 09 2008

Status

approved