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A209231
Number of binary words of length n such that there is at least one 0 and every run of consecutive 0's is of length >= 4.
4
0, 0, 0, 0, 1, 3, 6, 10, 15, 22, 33, 51, 80, 125, 193, 295, 449, 684, 1045, 1600, 2451, 3752, 5738, 8770, 13403, 20488, 31326, 47903, 73251, 112003, 171244, 261812, 400284, 612008, 935736, 1430709, 2187495, 3344566, 5113646, 7818463, 11953990, 18277013, 27944603
OFFSET
0,6
FORMULA
O.g.f.: x^4/((1-x)*(1-2*x+x^2-x^5)).
a(n) = Sum_{i=1..floor((n+1)/5)} binomial(n+1-3*i,2*i). - Andrew Howroyd, Sep 19 2025
EXAMPLE
a(5) = 3 because we have: {0,0,0,0,0}, {0,0,0,0,1}, {1,0,0,0,0}.
MATHEMATICA
nn=40; a=x^4/(1-x); CoefficientList[Series[(a+1)/((1-a x/(1-x)))*1/(1-x)-1/(1-x), {x, 0, nn}], x]
PROG
(PARI) a(n) = sum(i=1, (n+1)\5, binomial(n+1-3*i, 2*i)) \\ Andrew Howroyd, Sep 19 2025
CROSSREFS
Column 4 of A388547.
Sequence in context: A265071 A330910 A226239 * A137358 A143963 A385142
KEYWORD
nonn,easy
AUTHOR
Geoffrey Critzer, Jan 12 2013
EXTENSIONS
a(41) onwards from Andrew Howroyd, Sep 19 2025
STATUS
approved