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Numbers of length n binary words with fewer than 6 0-digits between any pair of consecutive 1-digits.
2

%I #8 Mar 13 2023 15:09:07

%S 1,2,4,8,16,32,64,128,255,507,1007,1999,3967,7871,15615,30976,61446,

%T 121886,241774,479582,951294,1886974,3742973,7424501,14727117,

%U 29212461,57945341,114939389,227991805,452240638,897056776,1779386436,3529560412,7001175484

%N Numbers of length n binary words with fewer than 6 0-digits between any pair of consecutive 1-digits.

%H Vincenzo Librandi, <a href="/A145114/b145114.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,0,0,0,0,-1,1).

%F G.f.: (1-x+x^7)/(1-3*x+2*x^2+x^7-x^8).

%e a(8) = 255 = 2^8-1, because 10000001 is the only binary word of length 8 with not less than 6 0-digits between any pair of consecutive 1-digits.

%p a:= n-> (Matrix([[2, 1$7]]). Matrix(8, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0$4, -1, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..35);

%t CoefficientList[Series[(1 - x + x^7) / (1 - 3 x + 2 x^2 + x^7 - x^8), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 06 2013 *)

%t LinearRecurrence[{3,-2,0,0,0,0,-1,1},{1,2,4,8,16,32,64,128},40] (* _Harvey P. Dale_, Mar 13 2023 *)

%Y 6th column of A145111.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Oct 02 2008