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A006980 Compositions: 6th column of A048004.

%I A006980 M1411 #36 May 22 2025 01:04:43
%S A006980 1,2,5,12,28,64,143,315,687,1485,3186,6792,14401,30391,63872,133751,
%T A006980 279177,581040,1206151,2497895,5161982,10646564,21919161,45052841,
%U A006980 92461171,189489255,387830160,792810956,1618840800,3301999647
%N A006980 Compositions: 6th column of A048004.
%C A006980 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
%D A006980 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006980 J. L. Yucas, Counting special sets of binary Lyndon words, Ars Combin., 31 (1991), 21-29.
%H A006980 Alois P. Heinz, <a href="/A006980/b006980.txt">Table of n, a(n) for n = 6..1000</a>
%H A006980 J. L. Yucas, <a href="/A006980/a006980.pdf">Counting special sets of binary Lyndon words</a>, Ars Combin., 31 (1991), 21-29. (Annotated scanned copy)
%H A006980 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,0,-1,-2,-4,-5,-4,-3,-2,-1).
%F A006980 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
%F A006980 G.f.: x^6 * (1-x)^2 / ((1-2*x+x^6) * (1-2*x+x^7)). - _Félix Balado_, May 20 2025
%p A006980 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
%o A006980 (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
%Y A006980 Cf. A048004, A006979, A000102, A000100, A000071.
%K A006980 nonn,easy
%O A006980 6,2
%A A006980 _Simon Plouffe_
%E A006980 Corrected definition: 6th column of A048004. - _Geoffrey Critzer_, Nov 09 2008