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A283834 Number of length-n binary vectors beginning with 0, ending with 1, and avoiding 4 consecutive 0's and 4 consecutive 1's.

%I A283834 #21 Feb 10 2023 05:46:47
%S A283834 1,0,1,2,4,6,12,22,41,74,137,252,464,852,1568,2884,5305,9756,17945,
%T A283834 33006,60708,111658,205372,377738,694769,1277878,2350385,4323032,
%U A283834 7951296,14624712,26899040,49475048,90998801,167372888,307846737,566218426,1041438052
%N A283834 Number of length-n binary vectors beginning with 0, ending with 1, and avoiding 4 consecutive 0's and 4 consecutive 1's.
%H A283834 Alois P. Heinz, <a href="/A283834/b283834.txt">Table of n, a(n) for n = 0..1000</a>
%H A283834 Stefano Bilotta, <a href="http://arxiv.org/abs/1605.03785">Variable-length Non-overlapping Codes</a>, arXiv preprint arXiv:1605.03785 [cs.IT], 2016 [See Table 2].
%H A283834 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,2,3,2,1).
%F A283834 G.f.: 1/((1+x)*(1+x^2)*(1-x-x^2-x^3)). - _Alois P. Heinz_, Mar 25 2017
%F A283834 a(n) = (1/4)*((-1)^n + i^n*(n+1 mod 2) - i^(n+3)*(n mod 2) + 2*A000073(n+2)). - _G. C. Greubel_, Feb 09 2023
%t A283834 CoefficientList[Series[1/((1+x)*(1+x^2)*(1-x-x^2-x^3)), {x,0,50}], x] (* _Indranil Ghosh_, Mar 26 2017 *)
%o A283834 (PARI) Vec(1/((1+x)*(1+x^2)*(1-x-x^2-x^3)) + O(x^50)) \\ _Indranil Ghosh_, Mar 26 2017
%o A283834 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x)*(1+x^2)*(1-x-x^2-x^3)) )); // _G. C. Greubel_, Feb 09 2023
%o A283834 (SageMath)
%o A283834 @CachedFunction
%o A283834 def b(n): # b = A000073
%o A283834     if (n<3): return (0,0,1)[n]
%o A283834     else: return b(n-1) + b(n-2) + b(n-3)
%o A283834 def A283834(n): return (1/4)*((-1)^n +i^n*((n+1)%2) -i^(n+3)*(n%2) +2*b(n+2))
%o A283834 [A283834(n) for n in range(41)] # _G. C. Greubel_, Feb 09 2023
%Y A283834 Cf. A000073, A094686, A283835, A283836, A283837, A283838.
%K A283834 nonn,easy
%O A283834 0,4
%A A283834 _N. J. A. Sloane_, Mar 25 2017
%E A283834 More terms from _Alois P. Heinz_, Mar 25 2017