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

%I A006979 M1410 #26 Nov 20 2023 15:05:06
%S A006979 0,0,0,0,0,1,2,5,12,28,63,139,303,653,1394,2953,6215,13008,27095,
%T A006979 56201,116143,239231,491326,1006420,2056633,4193706,8534653,17337764,
%U A006979 35162804,71205504,143990366,290795624,586566102,1181834852,2378701408
%N A006979 a(n) is the number of compositions of n in which the maximum part size is 5.
%C A006979 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
%D A006979 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006979 J. L. Yucas, Counting special sets of binary Lyndon words, Ars Combin., 31 (1991), 21-29.
%H A006979 Matthew House, <a href="/A006979/b006979.txt">Table of n, a(n) for n = 0..3390</a>
%H A006979 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,0,-1,-3,-4,-3,-2,-1).
%F A006979 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
%p A006979 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
%t A006979 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 *)
%Y A006979 Cf. A048003.
%K A006979 nonn
%O A006979 0,7
%A A006979 _Simon Plouffe_
%E A006979 More terms and better definition from _Alois P. Heinz_, Oct 29 2008
%E A006979 Offset corrected by _Matthew House_, Feb 11 2017