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%I #29 Feb 07 2024 09:36:33
%S 1,2,4,8,16,31,59,111,207,384,710,1310,2414,4445,8181,15053,27693,
%T 50942,93704,172356,317020,583099,1072495,1972635,3628251,6673404,
%U 12274314,22575994,41523738,76374073,140473833,258371673,475219609,874065146
%N From a problem in AI planning: a(n) = 4+a(n-1)+a(n-2)+a(n-3)+a(n-4)-a(n-5)-a(n-6)-a(n-7), n>7.
%C The number of length n binary words with fewer than 3 zeros between any pair of consecutive ones. - _Jeffrey Liese_, Dec 23 2010
%H Robert Israel, <a href="/A007800/b007800.txt">Table of n, a(n) for n = 1..3397</a>
%H T. Langley, J. Liese, and J. Remmel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Langley/langley2.html">Generating Functions for Wilf Equivalence Under Generalized Factor Order </a>, J. Int. Seq. 14 (2011) # 11.4.2
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,0,-1,1).
%F a(1)=1, a(2)=2, a(3)=4, a(4)=8, a(5)=16, a(n)=3*a(n-1)-2*a(n-2)+0*a(n-3)- a(n-4)+ a(n-5). - _Harvey P. Dale_, Apr 24 2013
%F G.f.: -x*(x^4-x+1) / ((x-1)^2*(x^3+x^2+x-1)). - _Colin Barker_, Aug 18 2014
%F 2*a(n) = A001590(n+4)-n. - _R. J. Mathar_, Aug 16 2017
%p for n from 1 to 5 do a[n]:= [1,2,4,8,16][n] od:
%p for n from 6 to 100 do a[n]:= 3*a[n-1]-2*a[n-2]-a[n-4]+a[n-5] od:
%p seq(a[n],n=1..100); # _Robert Israel_, Aug 19 2014
%t LinearRecurrence[{3,-2,0,-1,1},{1,2,4,8,16},40] (* _Harvey P. Dale_, Apr 24 2013 *)
%o (PARI) Vec(-x*(x^4-x+1)/((x-1)^2*(x^3+x^2+x-1)) + O(x^100)) \\ _Colin Barker_, Aug 18 2014
%Y Cf. A062544.
%K nonn,easy
%O 1,2
%A Peter Jonsson [ petej(AT)ida.liu.se ]
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