A006980 Compositions: 6th column of A048004.

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

Author

Simon Plouffe

Keywords

• nonn
• easy

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
• 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).

Crossrefs

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

Programs

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

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

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

Extensions

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